Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Chapter 6Calculus of Residues

An Introduction to Complex Analysis

Leonor Aquino-RuivivarMathematics Department

De La Salle University-Manila

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Isolated Singularities

A complex number z0 is called an isolated singularity ofa function f if there f fails to be analytic at z0 but thereexists a deleted neighborhood of z0 where f is analyticeverywhere.Examples:

The function f (z) =cos zz2 + 1

has isolated singularities at

z = i and z = −i .The function f (z) = sec z has infinitely many isolatedsingular points of the form zk = π

2 + kπ, k ∈ Z spreadout evenly along the real axis.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Isolated Singularities

A complex number z0 is called an isolated singularity ofa function f if there f fails to be analytic at z0 but thereexists a deleted neighborhood of z0 where f is analyticeverywhere.Examples:

The function f (z) =cos zz2 + 1

has isolated singularities at

z = i and z = −i .The function f (z) = sec z has infinitely many isolatedsingular points of the form zk = π

2 + kπ, k ∈ Z spreadout evenly along the real axis.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Isolated Singularities

A complex number z0 is called an isolated singularity ofa function f if there f fails to be analytic at z0 but thereexists a deleted neighborhood of z0 where f is analyticeverywhere.Examples:

The function f (z) =cos zz2 + 1

has isolated singularities at

z = i and z = −i .The function f (z) = sec z has infinitely many isolatedsingular points of the form zk = π

2 + kπ, k ∈ Z spreadout evenly along the real axis.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Isolated Singularities

A complex number z0 is called an isolated singularity ofa function f if there f fails to be analytic at z0 but thereexists a deleted neighborhood of z0 where f is analyticeverywhere.Examples:

The function f (z) =cos zz2 + 1

has isolated singularities at

z = i and z = −i .The function f (z) = sec z has infinitely many isolatedsingular points of the form zk = π

2 + kπ, k ∈ Z spreadout evenly along the real axis.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Isolated Singularities

A complex number z0 is called an isolated singularity ofa function f if there f fails to be analytic at z0 but thereexists a deleted neighborhood of z0 where f is analyticeverywhere.Examples:

The function f (z) =cos zz2 + 1

has isolated singularities at

z = i and z = −i .The function f (z) = sec z has infinitely many isolatedsingular points of the form zk = π

2 + kπ, k ∈ Z spreadout evenly along the real axis.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Principal Part of a Function

Definition: Let z0 be an isolated singular point of a function fwhich is analytic in the annulus D : 0 < |z − z0| < r where ris the distance from z0 to the nearest isolated singular pointof f other than z0. Then f has a Laurent seriesrepresentation of the form

f (z) =+∞∑

n=−∞an(z − z0)

n

for every z ∈ D. The terms of the series consisting of thenegative powers of (z − z0) form the principal part of f at theisolated singular point z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The function f (z) =sin zz2 has the Laurent series

f (z) =sin zz2 =

1z− z

3!+

z3

5!+ · · ·

in the annular domain |z| > 0 so the principal partconsists only of ds 1

z .

The function exp1z

has the Laurent series

exp1z

=∞∑

n=0

1n!zn

This shows that all except the first term of the seriesbelong to the principal part of f at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The function f (z) =sin zz2 has the Laurent series

f (z) =sin zz2 =

1z− z

3!+

z3

5!+ · · ·

in the annular domain |z| > 0 so the principal partconsists only of ds 1

z .

The function exp1z

has the Laurent series

exp1z

=∞∑

n=0

1n!zn

This shows that all except the first term of the seriesbelong to the principal part of f at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The function f (z) =sin zz2 has the Laurent series

f (z) =sin zz2 =

1z− z

3!+

z3

5!+ · · ·

in the annular domain |z| > 0 so the principal partconsists only of ds 1

z .

The function exp1z

has the Laurent series

exp1z

=∞∑

n=0

1n!zn

This shows that all except the first term of the seriesbelong to the principal part of f at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Classification of Singularities

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain 0 < |z − z0| < r , where r isthe distance from z0 to the closest singular point. Then z0 is

a removable singularity if the principal part of f at z0 isempty.an essential singularity if the principal part of f at z0contains infinitely nonzero terms.a pole of order k if a−k 6= 0 and a−n = 0 for all n > k .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Classification of Singularities

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain 0 < |z − z0| < r , where r isthe distance from z0 to the closest singular point. Then z0 is

a removable singularity if the principal part of f at z0 isempty.an essential singularity if the principal part of f at z0contains infinitely nonzero terms.a pole of order k if a−k 6= 0 and a−n = 0 for all n > k .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Classification of Singularities

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain 0 < |z − z0| < r , where r isthe distance from z0 to the closest singular point. Then z0 is

a removable singularity if the principal part of f at z0 isempty.an essential singularity if the principal part of f at z0contains infinitely nonzero terms.a pole of order k if a−k 6= 0 and a−n = 0 for all n > k .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Classification of Singularities

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain 0 < |z − z0| < r , where r isthe distance from z0 to the closest singular point. Then z0 is

a removable singularity if the principal part of f at z0 isempty.an essential singularity if the principal part of f at z0contains infinitely nonzero terms.a pole of order k if a−k 6= 0 and a−n = 0 for all n > k .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The functionsin zz2 has a pole of order 1 at 0.

The function exp1z

has an essential singularity at 0.

The functionsinh z

zhas a removable singularity at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The functionsin zz2 has a pole of order 1 at 0.

The function exp1z

has an essential singularity at 0.

The functionsinh z

zhas a removable singularity at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The functionsin zz2 has a pole of order 1 at 0.

The function exp1z

has an essential singularity at 0.

The functionsinh z

zhas a removable singularity at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

The functionsin zz2 has a pole of order 1 at 0.

The function exp1z

has an essential singularity at 0.

The functionsinh z

zhas a removable singularity at 0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues

Let z0 be an isolated singularity of a function f , which isanalytic in some annular domain D : 0 < |z − z0| < r .The coefficient a−1 of the Laurent series for f about z0in D is called the residue of f at z0, and is denoted byRes(f , z0).Examples:

The residue of f (z) =sin zz2 at 0 is 1.

The residue of f (z) =1− cos z

z2 at 0 is 0.

The residue of f (z) =ez

(z − 1)3 is e2 .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Remark

Since

a−1 =1

2πi

∫C

f (z) dz ⇒∫

Cf (z) dz = 2πia−1 = 2πi Res (f , z0)

for every positively oriented simple closed contour C insidethe annular domain D : 0 < |z − z0| < r .

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

The Residue Theorem

Theorem: Let f be analytic in an open set D containinga simple closed contour C and its interior, except at afinite number of isolated singular points z1, z2, . . . , zkinterior to C. Then∫

Cf (z) dz = 2πi

k∑i=1

Res(f , zi)

Remark: The residue is defined as the value of acontour integral, but when the singularity is either aremovable singularity or a pole, there are alternativeways of determining the value of the residue. Inparticular, a−1 = Res(f , z0) = 0 when z0 is a removablesingular point.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

The Residue Theorem

Theorem: Let f be analytic in an open set D containinga simple closed contour C and its interior, except at afinite number of isolated singular points z1, z2, . . . , zkinterior to C. Then∫

Cf (z) dz = 2πi

k∑i=1

Res(f , zi)

Remark: The residue is defined as the value of acontour integral, but when the singularity is either aremovable singularity or a pole, there are alternativeways of determining the value of the residue. Inparticular, a−1 = Res(f , z0) = 0 when z0 is a removablesingular point.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

The Residue Theorem

Theorem: Let f be analytic in an open set D containinga simple closed contour C and its interior, except at afinite number of isolated singular points z1, z2, . . . , zkinterior to C. Then∫

Cf (z) dz = 2πi

k∑i=1

Res(f , zi)

Remark: The residue is defined as the value of acontour integral, but when the singularity is either aremovable singularity or a pole, there are alternativeways of determining the value of the residue. Inparticular, a−1 = Res(f , z0) = 0 when z0 is a removablesingular point.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f , and let λ(z) = (z − z0)

mf (z), where m is apositive integer. Then f has a pole of order m at z0 ifand only if

λ has a removable singularity at z0; andlim

z→z0λ(z) 6= 0.

Example: Let f (z) =z3 + 1

(z − i)(z + 2)4 . Then f has a

simple pole at z1 = i and a pole of order 4 at z2 = −2.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f , and let λ(z) = (z − z0)

mf (z), where m is apositive integer. Then f has a pole of order m at z0 ifand only if

λ has a removable singularity at z0; andlim

z→z0λ(z) 6= 0.

Example: Let f (z) =z3 + 1

(z − i)(z + 2)4 . Then f has a

simple pole at z1 = i and a pole of order 4 at z2 = −2.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f , and let λ(z) = (z − z0)

mf (z), where m is apositive integer. Then f has a pole of order m at z0 ifand only if

λ has a removable singularity at z0; andlim

z→z0λ(z) 6= 0.

Example: Let f (z) =z3 + 1

(z − i)(z + 2)4 . Then f has a

simple pole at z1 = i and a pole of order 4 at z2 = −2.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f , and let λ(z) = (z − z0)

mf (z), where m is apositive integer. Then f has a pole of order m at z0 ifand only if

λ has a removable singularity at z0; andlim

z→z0λ(z) 6= 0.

Example: Let f (z) =z3 + 1

(z − i)(z + 2)4 . Then f has a

simple pole at z1 = i and a pole of order 4 at z2 = −2.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f , and let λ(z) = (z − z0)

mf (z), where m is apositive integer. Then f has a pole of order m at z0 ifand only if

λ has a removable singularity at z0; andlim

z→z0λ(z) 6= 0.

Example: Let f (z) =z3 + 1

(z − i)(z + 2)4 . Then f has a

simple pole at z1 = i and a pole of order 4 at z2 = −2.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f .

If z0 is a pole of order m, then Res

(f , z0) =1

(m − 1)!lim

z→z0

dm−1

dzm−1 [(z − z0)mf (z)].

If z0 is a simple pole, then Res(f , z0) = lim

z→z0(z − z0)f (z).

Example: Evaluate the residues at each of the singular

points of the function f (z) =z2 + 4z2 + 1

.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f .

If z0 is a pole of order m, then Res

(f , z0) =1

(m − 1)!lim

z→z0

dm−1

dzm−1 [(z − z0)mf (z)].

If z0 is a simple pole, then Res(f , z0) = lim

z→z0(z − z0)f (z).

Example: Evaluate the residues at each of the singular

points of the function f (z) =z2 + 4z2 + 1

.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f .

If z0 is a pole of order m, then Res

(f , z0) =1

(m − 1)!lim

z→z0

dm−1

dzm−1 [(z − z0)mf (z)].

If z0 is a simple pole, then Res(f , z0) = lim

z→z0(z − z0)f (z).

Example: Evaluate the residues at each of the singular

points of the function f (z) =z2 + 4z2 + 1

.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f .

If z0 is a pole of order m, then Res

(f , z0) =1

(m − 1)!lim

z→z0

dm−1

dzm−1 [(z − z0)mf (z)].

If z0 is a simple pole, then Res(f , z0) = lim

z→z0(z − z0)f (z).

Example: Evaluate the residues at each of the singular

points of the function f (z) =z2 + 4z2 + 1

.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let z0 be an isolated singular point of afunction f .

If z0 is a pole of order m, then Res

(f , z0) =1

(m − 1)!lim

z→z0

dm−1

dzm−1 [(z − z0)mf (z)].

If z0 is a simple pole, then Res(f , z0) = lim

z→z0(z − z0)f (z).

Example: Evaluate the residues at each of the singular

points of the function f (z) =z2 + 4z2 + 1

.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let f be defined by f (z) =p(z)

q(z), where p and

q are both analytic at z0 and p(z0) 6= 0. Then f has apole of order m at z0 if and only if z0 is a zero of orderm of q. If m = 1, then

Res(f , z0) =p(z0)

q′(z0)

Example: Find the residues of the function tanh z.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let f be defined by f (z) =p(z)

q(z), where p and

q are both analytic at z0 and p(z0) 6= 0. Then f has apole of order m at z0 if and only if z0 is a zero of orderm of q. If m = 1, then

Res(f , z0) =p(z0)

q′(z0)

Example: Find the residues of the function tanh z.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Residues at Poles

Theorem: Let f be defined by f (z) =p(z)

q(z), where p and

q are both analytic at z0 and p(z0) 6= 0. Then f has apole of order m at z0 if and only if z0 is a zero of orderm of q. If m = 1, then

Res(f , z0) =p(z0)

q′(z0)

Example: Find the residues of the function tanh z.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Applications

Use the residue theorem to evaluate the following integrals:

(a)∫

C

z + 4z2 − 3z − 10

dz, C : |z| = 6

(b)∫

C

z2 + 1z2 cos πz

, C : |z| = 0.6

(c)∫

C

eπz

(z2 + 1)2 , C : |z| =

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorem: Let C be a simple closed contour, and let d be adomain which contains C and its interior. If

f is analytic in D except for a finite number of poles interiorto C; andf (z) 6= 0 for every z ∈ C. Then

12πi

∫C

f ′(z)

f (z)dz = N − P

where N and P are the number of zeros and poles,respectively and counting multiplicities, of f interior to C.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorem: Let C be a simple closed contour, and let d be adomain which contains C and its interior. If

f is analytic in D except for a finite number of poles interiorto C; andf (z) 6= 0 for every z ∈ C. Then

12πi

∫C

f ′(z)

f (z)dz = N − P

where N and P are the number of zeros and poles,respectively and counting multiplicities, of f interior to C.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorem: Let C be a simple closed contour, and let d be adomain which contains C and its interior. If

f is analytic in D except for a finite number of poles interiorto C; andf (z) 6= 0 for every z ∈ C. Then

12πi

∫C

f ′(z)

f (z)dz = N − P

where N and P are the number of zeros and poles,respectively and counting multiplicities, of f interior to C.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorem: Let C be a simple closed contour, and let d be adomain which contains C and its interior. If

f is analytic in D except for a finite number of poles interiorto C; andf (z) 6= 0 for every z ∈ C. Then

12πi

∫C

f ′(z)

f (z)dz = N − P

where N and P are the number of zeros and poles,respectively and counting multiplicities, of f interior to C.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Rouche’s Theorem

Let C be a simple closed contour, and let D be adomain which contains C and its interior. Let f and g beanalytic in D, and let |g(z)| < |f (z)| for all z ∈ C. Thenf +g and f have the same number of zeros interior to C.Examples:

Show that every polynomial of degree n has exactly nzeros.If |a| > e, show that the equation azn − ez = 0 has nroots interior to the unit circle.Let D be a domain that contains the unit circle C and itsinterior. Let f be analytic in D and let |f (z)| < 1 for everyz ∈ C. Show that there is exactly one point z0 interior toC such that f (z0) = z0.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

We wish to evaluate integrals of the form∫ 2π

0F (cos θ, sin θ) dθ

where F is a rational function of sin θ and cos θ.The above integral is the parametrized form of some

contour integral∫

Cf (z) dz, where C is the positively

oriented unit circle.We may write∫ 2π

0F (cos θ, sin θ) dθ

=

∫C

F[

12

(z +

1z

),

12i

(z − 1

z

)]dziz

=

∫C

f (z) dz

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

We wish to evaluate integrals of the form∫ 2π

0F (cos θ, sin θ) dθ

where F is a rational function of sin θ and cos θ.The above integral is the parametrized form of some

contour integral∫

Cf (z) dz, where C is the positively

oriented unit circle.We may write∫ 2π

0F (cos θ, sin θ) dθ

=

∫C

F[

12

(z +

1z

),

12i

(z − 1

z

)]dziz

=

∫C

f (z) dz

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

We wish to evaluate integrals of the form∫ 2π

0F (cos θ, sin θ) dθ

where F is a rational function of sin θ and cos θ.The above integral is the parametrized form of some

contour integral∫

Cf (z) dz, where C is the positively

oriented unit circle.We may write∫ 2π

0F (cos θ, sin θ) dθ

=

∫C

F[

12

(z +

1z

),

12i

(z − 1

z

)]dziz

=

∫C

f (z) dz

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

We wish to evaluate integrals of the form∫ 2π

0F (cos θ, sin θ) dθ

where F is a rational function of sin θ and cos θ.The above integral is the parametrized form of some

contour integral∫

Cf (z) dz, where C is the positively

oriented unit circle.We may write∫ 2π

0F (cos θ, sin θ) dθ

=

∫C

F[

12

(z +

1z

),

12i

(z − 1

z

)]dziz

=

∫C

f (z) dz

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

where

f (z) =1iz

F[

12

(z +

1z

),

12i

(z − 1

z

)]Examples:

Evaluate the integral∫ 2π

0

dθ

2 + sin θusing the residue

theorem.Use the residue theorem to evaluate the integral∫ π

0

85 + 2 cos θ

dθ.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

where

f (z) =1iz

F[

12

(z +

1z

),

12i

(z − 1

z

)]Examples:

Evaluate the integral∫ 2π

0

dθ

2 + sin θusing the residue

theorem.Use the residue theorem to evaluate the integral∫ π

0

85 + 2 cos θ

dθ.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

where

f (z) =1iz

F[

12

(z +

1z

),

12i

(z − 1

z

)]Examples:

Evaluate the integral∫ 2π

0

dθ

2 + sin θusing the residue

theorem.Use the residue theorem to evaluate the integral∫ π

0

85 + 2 cos θ

dθ.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

where

f (z) =1iz

F[

12

(z +

1z

),

12i

(z − 1

z

)]Examples:

Evaluate the integral∫ 2π

0

dθ

2 + sin θusing the residue

theorem.Use the residue theorem to evaluate the integral∫ π

0

85 + 2 cos θ

dθ.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Trigonometric Integrals

where

f (z) =1iz

F[

12

(z +

1z

),

12i

(z − 1

z

)]Examples:

Evaluate the integral∫ 2π

0

dθ

2 + sin θusing the residue

theorem.Use the residue theorem to evaluate the integral∫ π

0

85 + 2 cos θ

dθ.

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Improper Integrals With Infinite Limits ofIntegration

Definition: Let p > 0 be a real number. A function f is

said to be of order1zp and denote this by

f (z) = O(

1zp

)if there exists a constant K > 0 such

that |f (z)| ≤ Kzp for sufficiently large |z|.

Example: Let f be the function defined by

f (z) =z2

(z2 + 1)(z2 + 4). Then f is of order O

(1z2

).

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Improper Integrals With Infinite Limits ofIntegration

Definition: Let p > 0 be a real number. A function f is

said to be of order1zp and denote this by

f (z) = O(

1zp

)if there exists a constant K > 0 such

that |f (z)| ≤ Kzp for sufficiently large |z|.

Example: Let f be the function defined by

f (z) =z2

(z2 + 1)(z2 + 4). Then f is of order O

(1z2

).

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Improper Integrals With Infinite Limits ofIntegration

Definition: Let p > 0 be a real number. A function f is

said to be of order1zp and denote this by

f (z) = O(

1zp

)if there exists a constant K > 0 such

that |f (z)| ≤ Kzp for sufficiently large |z|.

Example: Let f be the function defined by

f (z) =z2

(z2 + 1)(z2 + 4). Then f is of order O

(1z2

).

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

Theorem: If f is continuous along the real axis, and

f (z) = O(

1zp

)for p > 1, then

∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Corollary: If f (z) =p(z)

q(z)where p, q are relatively

prime polynomials, q has no real zeros, and deg q(z) ≥deg p(z) + 2, then∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

Theorem: If f is continuous along the real axis, and

f (z) = O(

1zp

)for p > 1, then

∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Corollary: If f (z) =p(z)

q(z)where p, q are relatively

prime polynomials, q has no real zeros, and deg q(z) ≥deg p(z) + 2, then∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

Theorem: If f is continuous along the real axis, and

f (z) = O(

1zp

)for p > 1, then

∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Corollary: If f (z) =p(z)

q(z)where p, q are relatively

prime polynomials, q has no real zeros, and deg q(z) ≥deg p(z) + 2, then∫ ∞

−∞f (x) dx = 2πi

k∑n=1

Res(f , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

If f is an even function, then∫ ∞

0f (x) dx =

12

∫ ∞

−∞f (x) dx .

Examples:∫ ∞

−∞

x2

(x2 + 1)(x2 + 4)dx∫ ∞

0

dxx6 + 64

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

If f is an even function, then∫ ∞

0f (x) dx =

12

∫ ∞

−∞f (x) dx .

Examples:∫ ∞

−∞

x2

(x2 + 1)(x2 + 4)dx∫ ∞

0

dxx6 + 64

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

If f is an even function, then∫ ∞

0f (x) dx =

12

∫ ∞

−∞f (x) dx .

Examples:∫ ∞

−∞

x2

(x2 + 1)(x2 + 4)dx∫ ∞

0

dxx6 + 64

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

If f is an even function, then∫ ∞

0f (x) dx =

12

∫ ∞

−∞f (x) dx .

Examples:∫ ∞

−∞

x2

(x2 + 1)(x2 + 4)dx∫ ∞

0

dxx6 + 64

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

If f is an even function, then∫ ∞

0f (x) dx =

12

∫ ∞

−∞f (x) dx .

Examples:∫ ∞

−∞

x2

(x2 + 1)(x2 + 4)dx∫ ∞

0

dxx6 + 64

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

If f (z) = O(

1zp

)for p > 0, then for every m > 0, then

limR→∞

∫CR

eimz f (z) dz = 0

If f (z) =p(z)

q(z)where p, q are relatively prime

polynomials, q has no real zeros, and deg q(z) > degp(z), then∫ ∞

−∞f (x)eimx dx = 2πi

k∑n=1

Res(f (z)eimz , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

If f (z) = O(

1zp

)for p > 0, then for every m > 0, then

limR→∞

∫CR

eimz f (z) dz = 0

If f (z) =p(z)

q(z)where p, q are relatively prime

polynomials, q has no real zeros, and deg q(z) > degp(z), then∫ ∞

−∞f (x)eimx dx = 2πi

k∑n=1

Res(f (z)eimz , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Theorems

Let f be a function which is analytic on the upper half planeexcept for a finite number of poles z1, z2, . . . , zk .

If f (z) = O(

1zp

)for p > 0, then for every m > 0, then

limR→∞

∫CR

eimz f (z) dz = 0

If f (z) =p(z)

q(z)where p, q are relatively prime

polynomials, q has no real zeros, and deg q(z) > degp(z), then∫ ∞

−∞f (x)eimx dx = 2πi

k∑n=1

Res(f (z)eimz , zn)

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

Evaluate the following improper integrals:∫ ∞

−∞

cos x(x2 + 4)(x2 + 9)

dx∫ ∞

0

x sin x(x2 + 1)(x2 + 4)

dx

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

Evaluate the following improper integrals:∫ ∞

−∞

cos x(x2 + 4)(x2 + 9)

dx∫ ∞

0

x sin x(x2 + 1)(x2 + 4)

dx

Mth643

Types ofSingularities

The ResidueTheorem

Rouche’sTheorem

Applicationsof theResidueTheorem

Examples

Evaluate the following improper integrals:∫ ∞

−∞

cos x(x2 + 4)(x2 + 9)

dx∫ ∞

0

x sin x(x2 + 1)(x2 + 4)

dx