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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Integral Theorems

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction

1. Much of the strength of complex analysis derives from the factthat the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero

, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contour

and in the contour.2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.

3. We will start by analyzing integrals across closed contours a bitmore carefully.

4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.

4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.

5. Then we will consider a few properties of domains that relate tothe Cauchy-Goursat Theorem.

6. The original motivation to investigate integrals over closedcontours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.

6. The original motivation to investigate integrals over closedcontours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics.

For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Introduction1. Much of the strength of complex analysis derives from the fact

that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.

2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit

more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to

the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed

contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.

The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D.

(That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.

1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)

2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.

3. For any closed contour in D we have that∫

Cf (z) dz = 0.

In the above situation, if C is a contour from z1 to z2, then∫C

f (z) dz = F(z2)−F(z1).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example.

The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.

An antiderivative of zn is1

n+1zn+1!

(Also see earlier presentation for the direct computation for the unitcircle.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.

An antiderivative of zn is1

n+1zn+1!

(Also see earlier presentation for the direct computation for the unitcircle.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.

An antiderivative of zn is1

n+1zn+1!

(Also see earlier presentation for the direct computation for the unitcircle.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.

An antiderivative of zn is1

n+1zn+1!

(Also see earlier presentation for the direct computation for the unitcircle.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example.

The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi

(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1.

Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero

(it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible

, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches)

, the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.

Also recall ∫C

z−1 dz =∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall

∫C

z−1 dz =∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz

=∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt

= 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of the function z−1 around the unit circle is2πi(?)

It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫

Cz−1 dz =

∫ 2π

0

(eit)−1

ieit dt

=∫ 2π

0i dt = 2πi

The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“1⇒3”).

If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫

Cf (z) dz = F

(z(b)

)−F(z(a)

)= F

(z(a)

)−F(z(a)

)= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫

Cf (z) dz

= F(z(b)

)−F(z(a)

)= F

(z(a)

)−F(z(a)

)= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫

Cf (z) dz = F

(z(b)

)−F(z(a)

)

= F(z(a)

)−F(z(a)

)= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫

Cf (z) dz = F

(z(b)

)−F(z(a)

)= F

(z(a)

)−F(z(a)

)

= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫

Cf (z) dz = F

(z(b)

)−F(z(a)

)= F

(z(a)

)−F(z(a)

)= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”).

Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2).

Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz

−∫

C2f (z) dz∫

C1f (z) dz =

∫C2

f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz

∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue

, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2

, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2

, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.

0 =∫

Cf (z) dz

=∫

C1+(−C2)f (z) dz

=∫

C1f (z) dz−

∫C2

f (z) dz∫C1

f (z) dz =∫

C2f (z) dz

Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”).

Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D.

For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D

(recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected)

define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ .

Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour

, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line.

We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f .

Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)

= limw→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ

= f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line

, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z)

, so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is

connected) define F(z) :=∫ z

z0

f (γ) dγ . Because the integral only

depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.

limw→z

F(w)−F(z)w− z

= limw→z

1w− z

(∫ w

z0

f (γ) dγ−∫ z

z0

f (γ) dγ

)= lim

w→z

1w− z

∫ w

zf (γ) dγ = f (z)

because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (finish).

Finally, the claimed equation follows from a resultfrom a previous presentation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (finish). Finally, the claimed equation follows from a resultfrom a previous presentation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof (finish). Finally, the claimed equation follows from a resultfrom a previous presentation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C. Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C.

Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C. Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C. Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C. Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior

of C. Then∫

Cf (z) dz = 0.

C

interior

exterior

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c.

Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a.

Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2

, mbc :=b+ c

2, and mca :=

c+a2

. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2.

Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

r

mab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

r

mbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbc

rmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcr

mca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz

=∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz

+∫

[b,mbc,mab,b]f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz

+∫

[mab,mbc,mca,mab]f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the

midpoints mab :=a+b

2, mbc :=

b+ c2

, and mca :=c+a

2. Then

ra

rb

rc

rmab

rmbcrmca

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a

, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b

,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c

, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣

so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)

and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.∫[a,b,c,a]

f (z) dz =∫

[a,mab,mca,a]f (z) dz+

∫[b,mbc,mab,b]

f (z) dz

+∫

[c,mca,mbc,b]f (z) dz+

∫[mab,mbc,mca,mab]

f (z) dz

implies that the absolute value of one of the integrals on the right is

greater than or equal to∣∣∣∣14∫

[a,b,c,a]f (z) dz

∣∣∣∣ . Thus, if a0 := a, b0 := b,

c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz

∣∣∣∣≥ 14

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣so that the lengths of the sides satisfy

l([a1,b1,c1,a1]

)≤ 1

2l([a0,b0,c0,a0]

)and so that the diameters satisfy

diam(∆(a1,b1,c1)

)≤ 1

2diam

(∆(a0,b0,c0)

).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1

rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2

rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2

qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2

qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2

qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2

qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3

qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3

qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3

qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3

qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3q

qq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qq

q∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq

∣∣∣∣∫[an,bn,cn,an]f (z) dz

∣∣∣∣ ≥ 14

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq

∣∣∣∣∫[an,bn,cn,an]f (z) dz

∣∣∣∣ ≥ 14

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq

∣∣∣∣∫[an,bn,cn,an]f (z) dz

∣∣∣∣ ≥ 14

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq

∣∣∣∣∫[an,bn,cn,an]f (z) dz

∣∣∣∣ ≥ 14

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq

∣∣∣∣∫[an,bn,cn,an]f (z) dz

∣∣∣∣ ≥ 14

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣

≥ ·· ·

≥ 14n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·

≥ 14n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣

l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)

≤ ·· · ≤ 12n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· ·

≤ 12n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)

diam(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)

≤·· · ≤ 12n diam

(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· ·

≤ 12n diam

(∆(a0,b0,c0)

)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

ra0

rb0

rc0

ra1

rb1rc1

ra2

rb2rc2 qa3

qb3

qc3qqq∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣ ≥ 1

4

∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz

∣∣∣∣ ≥ ·· ·≥ 1

4n

∣∣∣∣∫[a0,b0,c0,a0]f (z) dz

∣∣∣∣l([an,bn,cn,an]

)≤ 1

2l([an−1,bn−1,cn−1,an−1]

)≤ ·· · ≤ 1

2n l([a0,b0,c0,a0]

)diam

(∆(an,bn,cn)

)≤ 1

2diam

(∆(an−1,bn−1,cn−1)

)≤·· · ≤ 1

2n diam(∆(a0,b0,c0)

)Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0) =: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0. For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0)

=: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0. For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0) =: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0. For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0) =: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0. For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0) =: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0.

For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. By definition of the derivative

0 = limz→z0

f (z)− f (z0)z− z0

− f ′(z0) =: limz→z0

h(z)

Thus there is a function h so that for all z we have

f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)

and limz→z0

h(z) = 0. For any ε > 0 we can find an n so that

supz∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

l[a,b,c,a]diam(∆(a,b,c)

) .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣

≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣

≤ 4n∫

[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz

∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]

f (z) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣

= 4n∣∣∣∣∫[an,bn,cn,an]

h(z)(z− z0) dz∣∣∣∣ ≤ 4n

∫[an,bn,cn,an]

∣∣(z− z0)∣∣∣∣h(z)

∣∣ d|z|

≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

= 4n 12n l[a,b,c,a]

12n diam

(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣

≤ l[a,b,c,a]diam(∆(a,b,c)

)sup

z∈[an,bn,cn,an]

∣∣h(z)∣∣< ε

implies that∫

[a,b,c,a]f (z) dz = 0, because ε was arbitrary.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

t

t tt

t

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt

tt

t

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

t

t

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

t

ttt t t

ttt

t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

tt

tt t t

ttt

t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t

t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t

tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t t

ttt

t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

t

tt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

tt

t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

C

t

tt t

tt

ttt

t t tt

ttt

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes

1. The Cauchy-Goursat Theorem works as long as the function isanalytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically.

This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.

If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence.

But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is

analytic on a domain that contains the contour and the contour’sinterior.

2. But if C is the positively oriented unit circle, then∫C

z−1 dz = 2πi 6= 0 shows that the result need not hold when the

function is not analytic in the whole interior.

3. Side note:∫

Cz−2 dz = 0 shows that just because the function is

not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes

4. The problem with z−1 on the unit circle is apparently that thefunction is not analytic (not even defined) at z = 0.

5. Pictorially, the domain of z−1 has a “hole” at zero, and we wantto formalize that idea.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the

function is not analytic (not even defined) at z = 0.

5. Pictorially, the domain of z−1 has a “hole” at zero, and we wantto formalize that idea.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the

function is not analytic (not even defined) at z = 0.5. Pictorially, the domain of z−1 has a “hole” at zero

, and we wantto formalize that idea.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the

function is not analytic (not even defined) at z = 0.5. Pictorially, the domain of z−1 has a “hole” at zero, and we want

to formalize that idea.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition.

A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D

(remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected)

so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �

C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.

D

? �C

Simply connected means “no holes”.Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof.

For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem.

Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>

U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then for

any closed contour C in D we have∫

Cf (z) dz = 0.

Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.

-

6ℑ(z)

ℜ(z)

1

?

}

>

U

y

>U

)

Y

C

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof.

By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof. By the preceding theorem, integrals of f over closed contoursare zero.

By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.

Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition.

A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole

? �C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Definition. A domain D the is not simply connected is calledmultiply connected.

D

hole? �

C

Multiply connected means there are holes.Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫

Cf (z) dz+

n

∑j=1

∫Cj

f (z) dz = 0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a positively oriented simple closed contour.

LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫

Cf (z) dz+

n

∑j=1

∫Cj

f (z) dz = 0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C.

Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫

Cf (z) dz+

n

∑j=1

∫Cj

f (z) dz = 0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj.

Then∫C

f (z) dz+n

∑j=1

∫Cj

f (z) dz = 0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫

Cf (z) dz+

n

∑j=1

∫Cj

f (z) dz = 0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

r

C1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

�

�

�

-

-

1

]

�

C

-

�

?O

rC1

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrrrIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

r

rrrIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rr

rrIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrr

rIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrrr

I

R

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrrrI

R

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrrrIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Proof.

D

r-�

?O

�

�

�

-

-

1

]

�

C

C1

rrrrIR

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem.

Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2.

Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them.

Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof.

Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result

, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented

, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero.

Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between

them. Then∫

C1

f (z) dz =∫

C2

f (z) dz.

Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example.

The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.

We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.

We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.

We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi.

The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.

We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior

, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems

logo1

Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains

Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.

We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Integral Theorems