Complex Integration: Definite Integrals of Complex-Valued Functions and Contour Integrals, Lecture notes of Engineering Mathematics

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Eski¸sehir Technical University, Turkey

November 14, 2018

Today

Integrals

Integrals

Some examples

Example

If f (t) = u(t) + jv(t) is continuous in [a, b],

∫ (^) b a f^ (τ^ )dτ^ exists.

Example

Let f (t) = 2t + j 1 /t in t ∈ [2, 3]. Then, ∫ (^3)

2

(2τ + j 1 /τ )dτ =

2

2 τ dτ + j

2

τ

dτ

= τ 2

]τ = τ =2 +^ j^ ln^ τ^ ]

τ = τ = = 5 + j (ln 3 − ln 2)

Antiderivative

Let w(t) = u(t) + jv(t) and W (t) = U (t) + jV (t), where u, v, U and V are real-valued continuous functions on the interval a ≤ t ≤ b. Then, if W ′(t) = w(t) on the interval, then, U ′(t) = u(t) and V ′(t) = v(t).

A fundamental theorem of Calculus

Let W ′(t) = w(t), a ≤ t ≤ b, then, ∫ (^) b

a

w(τ )dτ = U (t)]bt=a + j V (t)]bt=a = W (b) − W (a)

Curves and Contours

Remark

Integrals of complex-valued functions of a complex variable are defined on curves.

Definition 2

A set of points z = (x, y) in the complex plane is said to be an arc if

x = x(t) y = y(t) a ≤ t ≤ b.

where x(t) and y(t) are cts.

Curves and Contours

Remark

Integrals of complex-valued functions of a complex variable are defined on curves.

Definition 2

A set of points z = (x, y) in the complex plane is said to be an arc if

x = x(t) y = y(t) a ≤ t ≤ b.

where x(t) and y(t) are cts. Let C be a set a points described as z(t) = x(t) + jy(t), a ≤ t ≤ b. where x(t) and y(t) are cts. Then the arc C is called a simple arc, if z(t 1 ) 6 = z(t 2 ) when t 1 6 = t 2. The arc C is called a simple closed curve, if z(t 1 ) 6 = z(t 2 ) when t 1 6 = t 2 , except z(b) = z(a).

Simple Curves

Definition

A curve z(t), a ≤ t ≤ b, is said to be a simple curve if z(t 1 ) 6 = z(t 2 ) when a ≤ t 1 < t 2 ≤ b except z(a) = z(b).

Definition

A curve z(t), a ≤ t ≤ b, is said to be a simple curve if z(t 1 ) 6 = z(t 2 ) when a ≤ t 1 < t 2 ≤ b except z(a) = z(b).

Definition

Let x(t) and y(t) be continuous functions of the real parameter t on a ≤ t ≤ b. Then, a set of points z = (x(t), y(t)) in the complex plane is said to be an arc. An arc C is a simple arc, if z(t 1 ) 6 = z(t 2 ). It is called a closed curve or Jordan curve whenever z(a) = z(b).

Parametric representation of Arc is not unique

Let z(t) be an arc defined as

z(t) = 2t^2 + j 2 t

Then, for t = φ(τ ), where φ(τ ) = −τ − 1, z(φ(τ )) is obtained as shown in the figure

(^0 1 2 3 4) Real 5 6 7 8 9

Imaginary

0

1

2

3

4

Example

Example

Find the length of the given simple arc

z(t) = cos(t) + j sin(t), 0 ≤ t ≤ π.

It is obvious, since it is a semi-circle.

z′(t) = − sin(t) + j cos(t), 0 ≤ t ≤ π.

Then, L =

∫ (^) π

0

|z′(τ )|dτ =

∫ (^) π

0

1 dτ = π.

Contour

Definition

Let z(t) = x(t) + jy(t), a ≤ t ≤ b. Then, it is called a smooth arc (or smooth curve, if a) z′(t) = x′(t) + jy′(t) exists and continuous on t ∈ [a, b], b) z′(t) 6 = 0, t ∈ (a, b). Note, z(t 1 ) 6 = z(t 2 ), where t 1 6 = t 2 , except t 1 = a and t 2 = b.

Definition

A contour (or piecewise smooth arc), is an arc consisting of finite number smooth arcs joined end to end.

Contour Integral

Definition

Suppose that z = z(t), t ∈ [a, b], represents a contour C and f (z) is piecewise continuous on C. Then, we define the line integral or contour integral of f as ∫

C

f (z)dz =

∫ (^) b

a

f (z(t)) z′(τ )dτ.

Note, since z(t) is assumed to be contour, its derivative exists and piecewise continuous on the interval, hence, the integral exists.

Example

x

y

O

− 2 j

2 j

Evaluate the integral (^) ∫

C

zdz,

on the contour shown in the figure.

Properties of the integrals

linearity

Let α and β be complex numbers, and f (z) and g(z) be (piecewise) continuous complex valued functions defined on a contour C, then, ∫

C

(αf (z) + βg(z)) dz = α

C

f (z)dz + β

C

g(z)dz.

Let C = C 1 ∪ C 2 be a contour such that it is a composition of two contours C 1 and C 2 , that is, if C is a contour from zo to z 1 , then, C 1 is a contour starting from zo to some zm and C 2 is a contour starting from zm to z 1. Furthermore, let f (z) be (piecewise) continuous complex valued function defined on a contour C, then, ∫

C

f (z)z = α

C 1

f (z)dz +

C 2

f (z)dz.

Let f (z) be (piecewise) continuous complex valued functions defined on a contour C, and, let −C corresponds to same contour in opposite direction, then, (^) ∫

−C

f (z)dz = −

C

f (z)dz.

Let f (z) be (piecewise) continuous complex valued functions defined on a contour C : z(t) = x(t) + jy(t), where a ≤ t ≤ b, then ∣ ∣∣ ∣

C

f (z)dz

C

|f (z)| dz ≤

C

|f (z)| |z′(t)|dz ≤ M L,

where M = maxz∈C |f (z)| and L is the length of the contour.