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Contour Integrals of Functions of a Complex Variable

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Introduction

• 1. Complex functions of a complex variable are usually integrated

along parametric curves.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Introduction

• 1. Complex functions of a complex variable are usually integrated

along parametric curves.

• 2. The integrals are ultimately reduced to integrals of complex

functions of a real variable as introduced in the previous presentation.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ②

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C

r

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C

r

• x(a),y(a)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C

r

• x(a),y(a)
• r

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A set C of points (x,y) in the complex plane is called an arc if and

• nly if there are continuous functions x(t) and y(t) with a ≤ t ≤ b so

that for every point (x,y) in C there is a t so that x = x(t) and y = y(t).

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C

r

• x(a),y(a)
• r
• x(b),y(b)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

With the usual switching between two-dimensional notation and complex notation, we also write the continuous function as z(t) = x(t)+iy(t).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is a Jordan arc or a simple arc if and only if for t1 = t2 we have z(t1) = z(t2).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is a Jordan arc or a simple arc if and only if for t1 = t2 we have z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is a Jordan arc or a simple arc if and only if for t1 = t2 we have z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■ ■

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■ ■ ✐

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■ ■ ✐ ②

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■ ■ ✐ ② ✙

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called a simple closed curve if, except for z(a) = z(b) we have that t1 = t2 implies z(t1) = z(t2).

✲ ✻

ℑ(z) ℜ(z)

q ✙ ■ ■ ✐ ② ✙ ✻

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction.

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction.

✲ ✻

ℑ(z) ℜ(z)

❑

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called smooth if and only if the function z(t) that traverses C is differentiable with continuous derivative and z′(t) = 0 for all t.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called smooth if and only if the function z(t) that traverses C is differentiable with continuous derivative and z′(t) = 0 for all t.

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ② ❃ ❯

C

r

• x(a),y(a)
• r
• x(b),y(b)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called smooth if and only if the function z(t) that traverses C is differentiable with continuous derivative and z′(t) = 0 for all t.

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ❃ ❯

C

• x(a),y(a)

r

r

• x(b),y(b)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc C is called smooth if and only if the function z(t) that traverses C is differentiable with continuous derivative and z′(t) = 0 for all t.

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ❃ ❯

C

• x(a),y(a)

r

r

• x(b),y(b)
• No!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point.

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ❃ ❯

C

• x(a),y(a)

r

r

• x(b),y(b)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definitions

An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point.

✲ ✻

ℑ(z) ℜ(z)

✶ ❄ ⑥ ❃ ❯ ❃ ❯

C

• x(a),y(a)

r

r

• x(b),y(b)
• O.k. for contours.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = eiθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a positively oriented simple closed curve.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = eiθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a positively oriented simple closed curve.

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = eiθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a positively oriented simple closed curve.

✲ ✻

ℑ(z) ℜ(z)

❑

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = e−iθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a simple closed curve, but it is not positively

• riented.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = e−iθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a simple closed curve, but it is not positively

• riented.

✲ ✻

ℑ(z) ℜ(z) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. With z(θ) = e−iθ and 0 ≤ θ ≤ 2π, the unit circle

C = {z ∈ C : |z| = 1} is a simple closed curve, but it is not positively

• riented.

✲ ✻

ℑ(z) ℜ(z)

❯

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. The length of a contour C parametrized by z(t) is

L :=

b

a

• z′(t)
• dt.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. The length of a contour C parametrized by z(t) is

L :=

b

a

• z′(t)
• dt.

Discussion.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. The length of a contour C parametrized by z(t) is

L :=

b

a

• z′(t)
• dt.

Discussion. L =

b

a

• z′(t)
• dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. The length of a contour C parametrized by z(t) is

L :=

b

a

• z′(t)
• dt.

Discussion. L =

b

a

• z′(t)
• dt

=

b

a

• x′(t)

2 +

• y′(t)

2 dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. The length of a contour C parametrized by z(t) is

L :=

b

a

• z′(t)
• dt.

Discussion. L =

b

a

• z′(t)
• dt

=

b

a

• x′(t)

2 +

• y′(t)

2 dt which is the length formula from multivariable calculus.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. Let f be a continuous function of a complex variable and

let C be a contour with parametrization z(t).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. Let f be a continuous function of a complex variable and

let C be a contour with parametrization z(t). Then we define the contour integral of f over C as

• C f(z) dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Definition. Let f be a continuous function of a complex variable and

let C be a contour with parametrization z(t). Then we define the contour integral of f over C as

• C f(z) dz :=

b

a f(z(t))z′(t) dt.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Note.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

β

α f(ξ(τ))ξ ′(τ) dτ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

β

α f(ξ(τ))ξ ′(τ) dτ

=

β

α f(z(φ(τ)))z′(φ(τ))φ ′(τ) dτ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

β

α f(ξ(τ))ξ ′(τ) dτ

=

β

α f(z(φ(τ)))z′(φ(τ))φ ′(τ) dτ

=

b

a f(z(t))z′(t) dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

β

α f(ξ(τ))ξ ′(τ) dτ

=

β

α f(z(φ(τ)))z′(φ(τ))φ ′(τ) dτ

=

b

a f(z(t))z′(t) dt

Therefore, the definition of the contour integral is sensible, as it only depends on the shape of the contour, not on the way we parametrize it.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Note. If φ is a differentiable function with continuous, nonzero

derivative that maps the interval [α,β] to the interval [a,b], then z(φ(τ)) is another parametrization of C. It just uses the parameter τ in [α,β] rather than the parameter t in [a,b]. But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ(τ) := z(φ(τ)) we have

β

α f(ξ(τ))ξ ′(τ) dτ

=

β

α f(z(φ(τ)))z′(φ(τ))φ ′(τ) dτ

=

b

a f(z(t))z′(t) dt

Therefore, the definition of the contour integral is sensible, as it only depends on the shape of the contour, not on the way we parametrize

• it. (Omitted proof that any two parametrizations “differ” by a φ as

above.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

=

−a

−b f(z(−τ)) d

dτ z(−τ) dτ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

=

−a

−b f(z(−τ)) d

dτ z(−τ) dτ =

−a

−b f(z(−τ))z′(−τ)(−1) dτ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

=

−a

−b f(z(−τ)) d

dτ z(−τ) dτ =

−a

−b f(z(−τ))z′(−τ)(−1) dτ

=

a

b f(z(t))z′(t) dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

=

−a

−b f(z(−τ)) d

dτ z(−τ) dτ =

−a

−b f(z(−τ))z′(−τ)(−1) dτ

=

a

b f(z(t))z′(t) dt = −

• C f(z) dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Rules.

• 1. If w is a complex number, then
• C wf(z) dz = w
• C f(z) dz.

2.

• C(f +g)(z) dz =
• C f(z) dz+
• C g(z) dz.
• 3. Let −C denote the same contour as C, only traversed in the
• pposite direction. Then z(−τ) with −b ≤ τ ≤ −a is a

parametrization and

• −C f(z) dz

=

−a

−b f(z(−τ)) d

dτ z(−τ) dτ =

−a

−b f(z(−τ))z′(−τ)(−1) dτ

=

a

b f(z(t))z′(t) dt = −

• C f(z) dz
• 4. If the endpoint of C1 is the starting point of C2, then the union of

the two contours in denoted C := C1 +C2 and we have

• C f(z) dz =
• C1

f(z) dz+

• C2

g(z) dz.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit

ieit dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit

ieit dt =

π

0 ieit dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit

ieit dt =

π

0 ieit dt

= eit

• π

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit

ieit dt =

π

0 ieit dt

= eit

• π

= eiπ −e0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Compute the contour integral of f(z) = |z| around the

upper half of the positively oriented unit circle.

• C f(z) dz

=

π

• eit

ieit dt =

π

0 ieit dt

= eit

• π

= eiπ −e0 = −2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Proof.

• C f(z) dz

=

b

a f

• z(t)
• z′(t) dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Proof.

• C f(z) dz

=

b

a f

• z(t)
• z′(t) dt

= F

• z(t)
• b

a

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Proof.

• C f(z) dz

=

b

a f

• z(t)
• z′(t) dt

= F

• z(t)
• b

a = F(z(b))−F(z(a))

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Theorem. If the complex function f(z) has an antiderivative F(z),

then the integral of f over the contour C parametrized with z(t), a ≤ t ≤ b is equal to

• C f(z) dz = F(z(b))−F(z(a)).

Proof.

• C f(z) dz

=

b

a f

• z(t)
• z′(t) dt

= F

• z(t)
• b

a = F(z(b))−F(z(a))

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt = 2πi

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt = 2πi In particular, for n an integer not equal to −1, the integral is zero.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt = 2πi In particular, for n an integer not equal to −1, the integral is zero. Note that the computation is not affected by the fact that the contour crosses a branch cut.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt = 2πi In particular, for n an integer not equal to −1, the integral is zero. Note that the computation is not affected by the fact that the contour crosses a branch cut. We choose a branch and stay consistent with it.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

logo1 Contours Contour Integrals Examples

• Example. Let n ∈ C(!) Integrate the function f(z) = zn around the

positively oriented unit circle.

• C zn dz

=

2π

• eitn ieit dt =

2π

iei(n+1)t dt = 1 n+1ei(n+1)t

• 2π

(n = −1)

• C z−1 dz

=

2π

i dt = 2πi In particular, for n an integer not equal to −1, the integral is zero. Note that the computation is not affected by the fact that the contour crosses a branch cut. We choose a branch and stay consistent with it. A value at a single point (where the power function is discontinuous) does not affect an integral.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable