Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

24.7 Find the real and imaginary parts of the functions (i)z^2 , (ii)ez, and (iii) coshπz.
By considering the values taken by these parts on the boundaries of the region
0 ≤x,y≤1, determine the solution of Laplace’s equation in that region that
satisfies the boundary conditions
φ(x,0) = 0,φ(0,y)=0,
φ(x,1) =x, φ(1,y)=y+sinπy.

24.8 Show that the transformation

w=

∫z

0

1

(ζ^3 −ζ)^1 /^2

dζ

transforms the upper half-plane into the interior of a square that has one corner

at the origin of thew-plane and sides of lengthL,where

L=

∫π/ 2

0

cosec^1 /^2 θdθ.

24.9 Thefundamental theorem of algebrastates that, for a complex polynomialpn(z)
of degreen, the equationpn(z) = 0 has preciselyncomplex roots. By applying
Liouville’s theorem (see the end of section 24.10) tof(z)=1/pn(z), prove that
pn(z) = 0 has at least one complex root. Factor out that root to obtainpn− 1 (z)
and, by repeating the process, prove the above theorem.
24.10 Show that, ifais a positive real constant, the function exp(iaz^2 ) is analytic and
→0as|z|→∞for 0<argz≤π/4. By applying Cauchy’s theorem to a suitable
contour prove that
∫∞

0

cos(ax^2 )dx=

√

π

8 a

.

24.11 The function

f(z)=(1−z^2 )^1 /^2

of the complex variablezis defined to be real and positive on the real axis in

the range− 1 <x<1. Using cuts running along the real axis for 1<x<+∞

and−∞<x<−1, show howf(z) is made single-valued and evaluate it on the

upper and lower sides of both cuts.

Use these results and a suitable contour in the complexz-plane to evaluate the

integral

I=

∫∞

1

dx

x(x^2 −1)^1 /^2

.

Confirm your answer by making the substitutionx=secθ.
24.12 By considering the real part of
∫
−izn−^1 dz
1 −a(z+z−^1 )+a^2

,

wherez=expiθandnis a non-negative integer, evaluate

∫π

0

cosnθ

1 − 2 acosθ+a^2

dθ

forareal and>1.
24.13 Prove that iff(z)hasasimplezeroatz 0 ,then1/f(z) has residue 1/f′(z 0 )there.
Hence evaluate ∫
π

−π

sinθ

a−sinθ

dθ,

whereais real and>1.