Chapter 4 Complex Integration - FCAMPENA...Chapter 4 Complex Integration An Introduction to Complex...

Mth643

Curves in C

ContourIntegration

The WindingNumber

Cauchy’sIntegralTheorem

The CauchyIntegralFormula

Consequencesof theCauchyIntegralFormula

Chapter 4Complex Integration

An Introduction to Complex Analysis

Leonor Aquino-RuivivarMathematics Department

De La Salle University-Manila

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Curves in C

ContourIntegration

The WindingNumber

Curves in C

Let I = [a, b] be a closed interval in the real line and let f bea complex-valued function defined in I.

Then the range of f is called a curve, a path or an arc Cin the complex plane.If f is continuous, then C is called a continuous curve,path or arc in the complex plane.f is called a parametric representation of the curve C.Conversely, given a curve C, a function f : I → C iscalled a parametric representation of C if C is the locusof all points in the range of f .

Mth643

Curves in C

ContourIntegration

The WindingNumber

Curves in C

Mth643

Curves in C

ContourIntegration

The WindingNumber

Curves in C

Mth643

Curves in C

ContourIntegration

The WindingNumber

Curves in C

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

1. Find the curve represented by the functionz = f (t) = 1 + 2eit , t ∈ [0, 2π].We have z − 1 = 2eit , so that |z − 1| = |2eit | = 2, whichshows that the curve is the circle with center at 1 andradius 2 oriented in the counterclockwise direction.

2. Let C be the square whose sides lie along the lines x = ±2and y = ±2, where the points are traced in thecounterclockwise direction.Then a parametricrepresentation of the square is given by

z = f (t) = x(t) + iy(t) =

−2 + it t ∈ [−2, 2]−t + 2i t ∈ [−2, 2]−2− it t ∈ [−2, 2]t − 2i t ∈ [−2, 2]

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

z = f (t) = x(t) + iy(t) =

−2 + it t ∈ [−2, 2]−t + 2i t ∈ [−2, 2]−2− it t ∈ [−2, 2]t − 2i t ∈ [−2, 2]

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

z = f (t) = x(t) + iy(t) =

−2 + it t ∈ [−2, 2]−t + 2i t ∈ [−2, 2]−2− it t ∈ [−2, 2]t − 2i t ∈ [−2, 2]

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

z = f (t) = x(t) + iy(t) =

−2 + it t ∈ [−2, 2]−t + 2i t ∈ [−2, 2]−2− it t ∈ [−2, 2]t − 2i t ∈ [−2, 2]

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Curves in C

ContourIntegration

The WindingNumber

Types of Curves

Let C be a continuous curve on the complex planeparametrized by z = f (t) = x(t) + iy(t), t ∈ I = [a, b]. Wesay that(a) C is a simple arc if f (t1) 6= f (t2) whenever t1 6= t2;(b) C is a simple closed curve if C\{ f (b) } is a simple

arc and f (a) = f (b);(c) C is said to be a smooth arc if f ′(t) = x ′(t) + iy ′(t) 6= 0

for all t ∈ I and both x ′(t) and y ′(t) are continuous in I;(d) C is called a contour or a piecewise smooth arc if C

is an arc consisting of a finite number of smooth arcsjoined end to end.

(e) C is called a simple closed contour if only the initialand final points z = f (a) and z = f (b) are identical.

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Curves in C

ContourIntegration

The WindingNumber

Types of Curves

Mth643

Curves in C

ContourIntegration

The WindingNumber

Types of Curves

Mth643

Curves in C

ContourIntegration

The WindingNumber

Types of Curves

Mth643

Curves in C

ContourIntegration

The WindingNumber

Types of Curves

Mth643

Curves in C

ContourIntegration

The WindingNumber

Types of Curves

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Curves in C

ContourIntegration

The WindingNumber

Inverse Contour

Let C be a contour defined by z = f (t), t ∈ I = [a, b].The inverse of C is the contour denoted by C which isparametrized by z = f (a + b − t). As t increases from ato b, a + b − t decreases from b to a, so C and Cconsist of exactly the same points but traversed inopposite directions. The inverse C may also beparametrized by z = f (−t), t ∈ [−b, −a].Example: Consider the contour C parametrized byz = eit , t ∈ [0, 2π] which is the unit circle traversed inthe counterclockwise direction. The inverse C of C isthe same circle traversed this time in the clockwisedirection, parametrized by z = ei(2π−t), t ∈ [0, 2π], orby z = e−it , t ∈ [−2π, 0] .

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Curves in C

ContourIntegration

The WindingNumber

Inverse Contour

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Curves in C

ContourIntegration

The WindingNumber

Inverse Contour

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Curves in C

ContourIntegration

The WindingNumber

Examples

Parametrize the following curves:

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Curves in C

ContourIntegration

The WindingNumber

Sum of Two Contours

Let C1 be a contour parametrized byz = f (t), t ∈ [a, b] and let C2 be a second contourparametrized by z = g(t), t ∈ [b, c], such thatf (b) = g(b). Then the sum C = C1 + C2 is the contourformed by the union of the two given contours and isparametrized by

z = h(t) =

{f (t) t ∈ [a, b]g(t) t ∈ [b, c]

Example: Let C1 be the contour parametrized byz = 2eit , t ∈ [0, π] and let C2 be the contourparametrized by z = 2eit , t ∈ [π, 2π]. Then C1 is theupper semicircle while C2 is the lower semicircle of thecircle of radius 2 with center at the origin, bothtraversed in the counterclockwise direction. C is thecircle traced in the counterclockwise direction.

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Curves in C

ContourIntegration

The WindingNumber

Sum of Two Contours

z = h(t) =

{f (t) t ∈ [a, b]g(t) t ∈ [b, c]

Mth643

Curves in C

ContourIntegration

The WindingNumber

Sum of Two Contours

z = h(t) =

{f (t) t ∈ [a, b]g(t) t ∈ [b, c]

Mth643

Curves in C

ContourIntegration

The WindingNumber

Definitions

Let f be a complex valued function of a real variable tdefined by

f (t) = u(t) + iv(t), t ∈ I = [a, b]

where u(t) and v(t) are both continuous in the intervalI. We can then define∫ b

af (t) dt =

au(t) dt + i

av(t) dt

Remark: Since the integral of a complex-valuedfunction of a real variable is defined in terms of theintegrals of its real and imaginary parts, it is clear thatthe properties of a definite integral may be extended tothe integral of complex-valued functions of a realvariable.Example: Evaluate

∫ 0−2 (1 + i)(cos t) dt .

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Curves in C

ContourIntegration

The WindingNumber

Definitions

f (t) = u(t) + iv(t), t ∈ I = [a, b]

af (t) dt =

au(t) dt + i

av(t) dt

∫ 0−2 (1 + i)(cos t) dt .

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Curves in C

ContourIntegration

The WindingNumber

Definitions

f (t) = u(t) + iv(t), t ∈ I = [a, b]

af (t) dt =

au(t) dt + i

av(t) dt

∫ 0−2 (1 + i)(cos t) dt .

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Curves in C

ContourIntegration

The WindingNumber

Definitions

f (t) = u(t) + iv(t), t ∈ I = [a, b]

af (t) dt =

au(t) dt + i

av(t) dt

∫ 0−2 (1 + i)(cos t) dt .

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Curves in C

ContourIntegration

The WindingNumber

Contour Integrals

Let C be a contour in the complex plane parametrizedby z = f (t), t ∈ [a, b], and let G(z) be acomplex-valued function which is continuous on thecontour C. Then(a) G(f (t)) is continuous on [a, b]

(b) dz =dzdt

dt = f ′(t) dt

The integral of G along the contour C is defined by∫C

G(z) dz =

aG(f (t) f ′(t) dt

Example: Evaluate the integral∫

dzz2 where C is the

circle |z| = 2 traversed in the counterclockwise directionand parametrized by z = f (t) = 2eit , t ∈ [0, 2π].The length L of a contour C defined by z = z(t) where

t ∈ [a, b] is given by L =

a|z ′(t)| dt .

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Curves in C

ContourIntegration

The WindingNumber

Contour Integrals

(b) dz =dzdt

dt = f ′(t) dt

G(z) dz =

dzz2 where C is the

a|z ′(t)| dt .

Mth643

Curves in C

ContourIntegration

The WindingNumber

Contour Integrals

(b) dz =dzdt

dt = f ′(t) dt

G(z) dz =

dzz2 where C is the

a|z ′(t)| dt .

Mth643

Curves in C

ContourIntegration

The WindingNumber

Contour Integrals

(b) dz =dzdt

dt = f ′(t) dt

G(z) dz =

dzz2 where C is the

a|z ′(t)| dt .

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Curves in C

ContourIntegration

The WindingNumber

Some Properties of Contour Integrals

Let F (z)and G(z) be continuous complex valued functionsdefined on a contour C, and let C be the inverse of C. Alsolet α be a complex constant. Then

(a)∫

CF (z)±G(z) dz =

F (z) dz ±∫

CG(z) dz

(b)∫

Cα F (z) dz = α

F (z) dz

(c)∫

CF (z) dz = −

F (z) dz

(d) If C = C1 + C2, then∫C

F (z) dz =

F (z) dz +

F (z) dz

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Curves in C

ContourIntegration

The WindingNumber

(a)∫

CF (z)±G(z) dz =

F (z) dz ±∫

CG(z) dz

(b)∫

Cα F (z) dz = α

F (z) dz

(c)∫

CF (z) dz = −

F (z) dz

F (z) dz =

F (z) dz +

F (z) dz

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Curves in C

ContourIntegration

The WindingNumber

(a)∫

CF (z)±G(z) dz =

F (z) dz ±∫

CG(z) dz

(b)∫

Cα F (z) dz = α

F (z) dz

(c)∫

CF (z) dz = −

F (z) dz

F (z) dz =

F (z) dz +

F (z) dz

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Curves in C

ContourIntegration

The WindingNumber

(a)∫

CF (z)±G(z) dz =

F (z) dz ±∫

CG(z) dz

(b)∫

Cα F (z) dz = α

F (z) dz

(c)∫

CF (z) dz = −

F (z) dz

F (z) dz =

F (z) dz +

F (z) dz

Mth643

Curves in C

ContourIntegration

The WindingNumber

(a)∫

CF (z)±G(z) dz =

F (z) dz ±∫

CG(z) dz

(b)∫

Cα F (z) dz = α

F (z) dz

(c)∫

CF (z) dz = −

F (z) dz

F (z) dz =

F (z) dz +

F (z) dz

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Curves in C

ContourIntegration

The WindingNumber

(e) Theorem:Let G be a complex-valued function which iscontinuous on a contourC parametrized byz = f (t), t ∈ [a, b], and let L be the length of thiscontour. Let G be bounded on C, so there exists M > 0such that |G(z)| ≤ M for every z ∈ C. Then∣∣∣∣∫

CG(z) dz

∣∣∣∣ ≤ ML

(f) Example: Without evaluating the integral, show that∣∣∣∣∫C

dzz2 − 1

∣∣∣∣ ≤ 4π

where C is the circle |z| = 2 traversed in thecounterclockwise direction.

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Curves in C

ContourIntegration

The WindingNumber

CG(z) dz

∣∣∣∣ ≤ ML

dzz2 − 1

∣∣∣∣ ≤ 4π

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Curves in C

ContourIntegration

The WindingNumber

CG(z) dz

∣∣∣∣ ≤ ML

dzz2 − 1

∣∣∣∣ ≤ 4π

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Curves in C

ContourIntegration

The WindingNumber

Exercises

1. Evaluate∫

Cz dz, C : z(t) = 2eit ,−π

2≤ t ≤ π

22. Evaluate

∫C (z − z0)

n dz, C : |z − z0| = r > 0 traversedcounterclockwise once, n is a positive integer.

3. Show that∣∣∣∣∫

∣∣∣∣ ≤ k , where n is a non-negative

integer, k > 0 and C is the path from z = i to z = k + i .

Mth643

Curves in C

ContourIntegration

The WindingNumber

Variation of the Logarithm Along a Contour

Let C : z = z(t), t ∈ [a, b] be a contour contained in adomain D. Let G be a function which is analytic in D andnon-vanishing in C. Let φ be a function continuous on [a, b]such that

eφ(t) = G(z(t))

for each t ∈ [a, b], so that φ(t) is one of the values oflog[G(z(t))].

The variation of log G(z(t)) along the contour C is thenumber

4C log G(z) = φ(b)− φ(a)

The function φ is given by

φ(T ) =

G′(z(t)z ′(t)G(z(t))

dt+k , a ≤ T ≤ b, ek = G(z(a))

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Curves in C

ContourIntegration

The WindingNumber

eφ(t) = G(z(t))

φ(T ) =

dt+k , a ≤ T ≤ b, ek = G(z(a))

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Curves in C

ContourIntegration

The WindingNumber

eφ(t) = G(z(t))

φ(T ) =

dt+k , a ≤ T ≤ b, ek = G(z(a))

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Curves in C

ContourIntegration

The WindingNumber

4C log G(z) = φ(b)− φ(a) =

G′(z)

G(z)dz

If C is a closed contour, then

eφ(a) = G(z(a)) = G(z(b)) = eφ(b) ⇒ φ(b)− φ(a)

This shows that

4C log G(z) =1

2πi4C log G(z) = k

is an integer.

Example:If f (z) = z2, and C : z(t) = eit , 0 ≤ t ≤ π

2, find

4C log f (z).Example: Let C be the unit circle oriented

counterclockwise. If f (z) = z2, find1

2πi4C log f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

4C log G(z) = φ(b)− φ(a) =

G′(z)

G(z)dz

This shows that

4C log G(z) =1

2πi4C log G(z) = k

is an integer.

2, find

2πi4C log f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

4C log G(z) = φ(b)− φ(a) =

G′(z)

G(z)dz

This shows that

4C log G(z) =1

2πi4C log G(z) = k

is an integer.

2, find

2πi4C log f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

4C log G(z) = φ(b)− φ(a) =

G′(z)

G(z)dz

This shows that

4C log G(z) =1

2πi4C log G(z) = k

is an integer.

2, find

2πi4C log f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

4C log G(z) = φ(b)− φ(a) =

G′(z)

G(z)dz

This shows that

4C log G(z) =1

2πi4C log G(z) = k

is an integer.

2, find

2πi4C log f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

The Winding Number

C be a contour, and let G(z) = z − s, where s 6∈ C. Then Gis analytic in any domain that contains C and isnon-vanishing on C.

Hence,

∆C log G(z) =1

dzz − s

depends only on C and s, and will be denoted byν(C, s).If C is a closed contour, then ν(C, s) is an integer calledthe winding number of C with respect to s.The winding number is the number of times the closedcontour C winds around s along the counterclockwisedirection.

Mth643

Curves in C

ContourIntegration

The WindingNumber

The Winding Number

Hence,

∆C log G(z) =1

dzz − s

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Curves in C

ContourIntegration

The WindingNumber

The Winding Number

Hence,

∆C log G(z) =1

dzz − s

Mth643

Curves in C

ContourIntegration

The WindingNumber

The Winding Number

Hence,

∆C log G(z) =1

dzz − s

Mth643

Curves in C

ContourIntegration

The WindingNumber

Example

ν(C, z1) = 2ν(C, z2) = 1

Mth643

Curves in C

ContourIntegration

The WindingNumber

Example

ν(C, z1) = 2ν(C, z2) = 1

Mth643

Curves in C

ContourIntegration

The WindingNumber

Example

ν(C, z1) = 2ν(C, z2) = 1

Mth643

Curves in C

ContourIntegration

The WindingNumber

Properties of the Winding Number

Let C be a contour. Then ν(C, s) is a continuousfunction of s not on C.If C is a closed contour and S is a connected set thatdoes not intersect C, then ν(C, s) is constant in S.Lemma: Let C be a closed contour, and let S be aconnected set disjoint from C. If there exists asequence (sn) in S such that sn →∞ as n →∞, thenν(C, s) = 0 for each s ∈ S.If C is a simple closed contour , then ν(C, s) = 0 forevery s exterior to C.Let C be a simple closed contour. Then ν(C, ξ) = ±1for all ξ interior to C.

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Simply Connected Domains

Let D be a domain. D is said to be simply connected if everysimple closed contour C which lies completely inside Cencloses only points of D, i.e. the interior of C is a subset ofD. Otherwise, D is said to be a multiply-connected domain.

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

In each of the following, determine whether D issimply-connected or multiply-connected.(a) D is the region between the square with sides along the

lines x = ±1 and y = ±1 and the circle |z| = 3,exclusive.

(b) D = { z ∈ C : |Arg z| < π

(c) D = { z = x + iy : y > x2 }(d) D is the region interior to the circle C : |z| = 6 but

exterior to each of the circlesC1 : |z − 3| = 1, C2 : |z + 3i | = 1 and C3 : |z + 2i | = 2.

(e) D = C\{ z ∈ C : Re z ≤ 0, Im z = 0 }(f) D = C\{ 0 }

(g) D is the horizontal strip |Im z| < 1(h) D is the annulus 1 < |z| < 2

Mth643

Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

Lemma: Let f be continuous in a domain D and let Fbe analytic in D such that

F ′(z) = f (z) for every z ∈ D

If Γ is a contour which lies completely inside D with z1as initial point and z2 as its terminal point, then∫

Γf (z) dz = F (z2)− F (z1)

Remark: The preceding lemma says that if F is anantiderivative of f and F is analytic in a domain D, thenthe integral of f in D over contours that lie in D isindependent of the contour chosen, since its valuedepends only on the initial and the terminal points ofthe contour, and not the contour itself.

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Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

Γf (z) dz = F (z2)− F (z1)

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Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

Γf (z) dz = F (z2)− F (z1)

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Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

Lemma: Let F be analytic in a domain D and let f becontinuous in D with F ′(z) = f (z) for each z ∈ D. If C1and C2 are two contours lying in D with the same initialand the same terminal points. Then∫

f (z) dz =

f (z) dz

Lemma: (Goursat’s Lemma) Let f be analytic in adomain D that contains a rectangle R. If C is theoriented boundary of R, then∫

Cf (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

f (z) dz =

f (z) dz

Cf (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Some Preliminaries

f (z) dz =

f (z) dz

Cf (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

(a) Evaluate∫

Csin z dz where

C : z(t) = 2eit , −π

2≤ t ≤ π

(b) Let f (z) = ez and let C be the square whose sides lielong the lines x = ±2 and y = ±2. Then∫

Cez dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

(a) Evaluate∫

Csin z dz where

C : z(t) = 2eit , −π

2≤ t ≤ π

Cez dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Examples

(a) Evaluate∫

Csin z dz where

C : z(t) = 2eit , −π

2≤ t ≤ π

Cez dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Theorem: Let D be any open set, and let C be a closedcontour in D such that ν(C, s) = 0 for every s 6∈ D.Then for every function f which is analytic in D, we have∫

Cf (z) dz = 0

Example: Let D = { z = x + iy ∈ C : x > 0, y > 0 }and let C = |z − (2 + 2i)| = 1. Then nu(C, s) = 0 foreach s not in D. Hence, for every function f analytic in

D,we have∫

Cf (z) dz = 0.

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Curves in C

ContourIntegration

The WindingNumber

Cf (z) dz = 0

D,we have∫

Cf (z) dz = 0.

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Curves in C

ContourIntegration

The WindingNumber

Cf (z) dz = 0

D,we have∫

Cf (z) dz = 0.

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Curves in C

ContourIntegration

The WindingNumber

The Cauchy Integral Theorems

(Cauchy Integral Theorem for Simply ConnectedDomains): Let f be analytic in a simply connecteddomain D, and let C be any closed contour in D. Then∫

Cf (z) dz = 0

(Cauchy Integral Theorem for Simple ClosedContours):Let C be a simple closed contour, and let fbe analytic in an open set D that contains C and itsinterior. Then ∫

Cf (z) dz = 0

(Cauchy Integral Theorem for Mulitply-ConnectedDomains): Let C, C1, C2, . . . , Cn be simple closedcontours oriented in the positive direction such thateach Cj is interior to C and exterior to Ck wheneverj 6= k . Let

Γ = C − C1 − C2 − · · · − Cn = C + C1 + C2 + · · ·+ Cn

Let R be the region consisting of the contoursC, C1, . . . , Cn and the points interior to C but exterior toeach Cj , so that the interior of R is a multiply-connecteddomain. Let D be an open set containing R. Then∫

Cf (z) dz =

f (z) dz+

f (z) dz+· · ·+∫

f (z) dz

Equivalently, ∫Γ

f (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Cf (z) dz = 0

Γ = C − C1 − C2 − · · · − Cn = C + C1 + C2 + · · ·+ Cn

Cf (z) dz =

f (z) dz+

f (z) dz+· · ·+∫

f (z) dz

Equivalently, ∫Γ

f (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Cf (z) dz = 0

Γ = C − C1 − C2 − · · · − Cn = C + C1 + C2 + · · ·+ Cn

Cf (z) dz =

f (z) dz+

f (z) dz+· · ·+∫

f (z) dz

Equivalently, ∫Γ

f (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Cf (z) dz = 0

Γ = C − C1 − C2 − · · · − Cn = C + C1 + C2 + · · ·+ Cn

Cf (z) dz =

f (z) dz+

f (z) dz+· · ·+∫

f (z) dz

Equivalently, ∫Γ

f (z) dz = 0

Mth643

Curves in C

ContourIntegration

The WindingNumber

Exercises

1. Evaluate the following integrals:

(a)∫ 2πi

iez dz, where the contour of integration is any

contour joining z = i to z = 2πi .

(b)∫ πi

0cosh z dz, over any contour joining z = 0 to z =

2. Evaluate∫

dzz2 + 1

for each of the following contours:

(a) Γ is the positively oriented boundary of the regionbetween the circles |z − i | = 1/2 and |z − i | = 3/2

(b) Γ is the union of the segments joining 0 to 2 to 2 + i to 03. Let C : |z| = 1 be traversed once counterclockwise.

Evaluate the following integrals:

(a)∫

z2 + 1cosh z

(b)∫

sinh zcos2 z

(c) ∫C

cosh 2z + sinh(z/2)

(z2 − 4z + 5)(z2 − 8z + 8)

Mth643

Curves in C

ContourIntegration

The WindingNumber

Indefinite Integrals

Let f be analytic in a domain D, let z0 be a fixed point inD, and let z be a variable point in D. If the integral∫ z

z0f (ξ)dξ is taking along a contour that lies completely

in D, then the integral is independent of path, and maybe treated as a function of the variable point z.Then the function

F0(z) =

f (ξ)dξ

is analytic in D, and F ′0(z) = f (z).

Mth643

Curves in C

ContourIntegration

The WindingNumber

The Cauchy Integral Formula

Theorem: (Cauchy Integral Formula) Let C be asimple closed contour and let f be analytic in an openset D which contains C and its interior. If z0 is any pointinterior to C, then

f (z0) =1

z − z0dz

where C is oriented in the positive direction.

Example: Evaluate∫

2z + 2

− 5z − 2

dz, C : |z| = 3.

cos zz2 + 4

dz, : |z − i | = 2.

Mth643

Curves in C

ContourIntegration

The WindingNumber

f (z0) =1

z − z0dz

2z + 2

− 5z − 2

dz, C : |z| = 3.

cos zz2 + 4

dz, : |z − i | = 2.

Mth643

Curves in C

ContourIntegration

The WindingNumber

f (z0) =1

z − z0dz

2z + 2

− 5z − 2

dz, C : |z| = 3.

cos zz2 + 4

dz, : |z − i | = 2.

Mth643

Curves in C

ContourIntegration

The WindingNumber

f (z0) =1

z − z0dz

2z + 2

− 5z − 2

dz, C : |z| = 3.

cos zz2 + 4

dz, : |z − i | = 2.

Mth643

Curves in C

ContourIntegration

The WindingNumber

Derivatives of Analytic Functions

Theorem: (Generalized Cauchy Integral Formula) LetC be a simple closed contour, and let f be analytic in anopen set D containing C and its interior. If z0 is interiorto C, then

f (n)(z0) =n!

(z − z0)n+1 dz (1)

where C is oriented in the positive direction andn = 0, 1, 2, . . ..

dzz2 , C : |z| = 2

(Derivatives of analytic functions):Let f be analytic in an open set D. Then all the derivatives of f exist and areanalytic functions in D.

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ContourIntegration

The WindingNumber

f (n)(z0) =n!

(z − z0)n+1 dz (1)

dzz2 , C : |z| = 2

Mth643

Curves in C

ContourIntegration

The WindingNumber

f (n)(z0) =n!

(z − z0)n+1 dz (1)

dzz2 , C : |z| = 2

Mth643

Curves in C

ContourIntegration

The WindingNumber

f (n)(z0) =n!

(z − z0)n+1 dz (1)

dzz2 , C : |z| = 2

Mth643

Curves in C

ContourIntegration

The WindingNumber

Exercises

1. Let C be the square whose sides lie along the linesx = ±7 and y = ±7, oriented counterclockwise.Evaluate the following integrals:

(a)∫

sin z(z − iπ)5 dz

(b)∫

(z + i)4 dz

(c)∫

(cos z + sin z)2

(z − iπ/4)2

2. Let f be analytic on and inside a simple closed contour Cwhich is oriented positively. If z0 is not on C, verify that∫

f ′(z)

z − z0dz =

(z − z0)2 dz

3. Use Cauchy’s generalized integral formula to show that iff is analytic on and inside |z − z0| = r , then

f (n)(z0) =n!

∫ 2π

0f (z0 + reiθ)e−inθ dθ

Mth643

Curves in C

ContourIntegration

The WindingNumber

Consequences of the Cauchy Integral Formula

(Morera’s Theorem): Let f be continuous in a simplyconnected domain D. If

∫C f (z) dz = 0 for any closed

contour C which lies inside D, then f is analytic in D.(Cauchy’s Inequality): Let C be the positively orientedcircle |z − z0| = r and let f be analytic in an open set Dcontaining C and its interior. If M is the least upperbound of |f (z)| on C, then

|f (n)(z0)| ≤Mn!

rn , n = 0, 1, 2, . . .

(Liouville): If f is bounded and entire, then f is aconstant function.(Fundamental Theorem of Algebra): Let p(z) be apolynomial function of degree n > 1. Then the equationf (z) = 0 has at least one root.

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Curves in C

ContourIntegration

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|f (n)(z0)| ≤Mn!

rn , n = 0, 1, 2, . . .

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Curves in C

ContourIntegration

The WindingNumber

|f (n)(z0)| ≤Mn!

rn , n = 0, 1, 2, . . .

Mth643

Curves in C

ContourIntegration

The WindingNumber

|f (n)(z0)| ≤Mn!

rn , n = 0, 1, 2, . . .

Mth643

Curves in C

ContourIntegration

The WindingNumber

|f (n)(z0)| ≤Mn!

rn , n = 0, 1, 2, . . .

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

Mth643

Curves in C

ContourIntegration

The WindingNumber

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