Complex Integration and Analytic Functions, Lecture notes of Engineering Science and Technology

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MAT 247 Engineering Mathematics

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

November 19, 2018

Today

Integrals

Sequences and Series in complex plane

Line integrals

Complex-valued functions are not necessarily to be defined as function of a real parameter t.

Line integrals

Complex-valued functions are not necessarily to be defined as function of a real parameter t. Integration of a complex function f (z) with a complex variable z can be done by line integral.

Let z := z(t) = x(t) + jy(t) (a ≤ t ≤ b), be a contour C, extending from z 1 = z(a) to z 2 = z(b). Furthermore, let f (z) = f (z(t)) = u(x(t), y(t)) + jv(x(t), y(t)), be (piecewise) continuous complex valued function on the interval a ≤ t ≤ b. Then, since ∫

C

f (z)dz =

∫ (^) b

a

(f (z(t))) z′(t)dt,

∫ (^) b

a

(f (z(t))) z′(t)dt =

∫ (^) b

a

(u(x(t), y(t)) + jv(x(t), y(t)))

x′(t) + jy′(t)

dt.

By abuse of notation, drop t parameter, ∫ (^) b

a

f z′dt =

∫ (^) b

a

ux′^ − vy′

dt + j

∫ (^) b

a

vx′^ + uy

dt.

Cauchy-Goursat Theorem

Theorem

If a function f is analytic at all points interior to and on a simple closed contour (^) ∫

C

f (z)dz = 0,

Examples

Let C be a unit circle and f (z) = (^) (z 2 z+4). Then, evaluate the following integral (^) ∫

C

f (z) z − 1 / 2 j

dz.

Examples

Let C be a unit circle and f (z) = (^) (z 2 z+4). Then, evaluate the following integral (^) ∫

C

f (z) z − 1 / 2 j

dz.

Let C be a unit circle, then, evaluate the following integral ∫

C

z^10 + (1 − z)^100 (z^4 − 5)z dz.

Theorem

If a function f is continuous on a domain D and if ∫

C

f (z)dz = 0,

for every closed contour lying in D, then, f is analytic throughout D.

Theorem

If a function f is not constant in a domain, then, |f (z)| has no maximum value in that domain.

Theorem

If a function f is entire and bounded for all values of z in the complex plane, then, f (z) is constant throughout the plane.

Corollary

Suppose that a function f is continuous on a closed bounded region D and that it is analytic and not constant in the interior of D. Then, f (z) achieves its maximum on the boundary of D.

Fundamental theorem of algebra

Theorem

Any polynomial

P (z) = ao + a 1 z + a 2 z^2 +... + anzn, an 6 = 0,

of degree n ≥ 1 has at least one zero.

Convergence of Series

An infinite series of complex numbers

∑^ ∞

n=

zn = z 1 + z 2 +... + zn +... ,

is said to converge to a number S, which corresponds to sum of the series, if the sequence of partial sums, which means

SN =

∑^ N

n=

zn,

converges to a number S, then, we write,

∑^ ∞

n=

zn = S.

Theorem

Suppose that zn = xn + jyn, n = 1, 2 ,... , n,... and S = X + jY. Then,

∑^ ∞

n=

zn = S,

if and only if ∑∞

n=

xn = X ;

∑^ ∞

n=

yn = Y.

Taylor Series

Taylor series expansion of f about zo is called Maclaurin series, that is,

f (z) = f (0) + f ′(0) 1!

(z) + f ′′(0) 2!

(z)^2 +... (|z| < Ro)

Example

What is the domain of an entire function that can be represented as a Taylor series expansion?

Example

What is the Maclaurin series expansion of sin(z)?

Examples

Examples

z − 1

∑^ ∞

n=

zn,

z + 1

∑^ ∞

n=

(−1)nzn,

I (^) What is the Taylor series expansion of (^) z1+ (^3) +zz 2