Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.14 Exercises

We have seen that

∫

Γand

∫

γvanish, and if we denotezbyxalong the lineABthen it
has the valuez=xexp 2πialong the lineDC(note that exp 2πimust not be set equal to
1 until after the substitution forzhas been made in

∫

DC). Substituting these expressions,

∫∞

0

dx

(x+a)^3 x^1 /^2

+

∫ 0

∞

dx

[xexp 2πi+a]^3 x^1 /^2 exp(^122 πi)

=

3 π

4 a^5 /^2

.

Thus (

1 −

1

expπi

)∫∞

0

dx

(x+a)^3 x^1 /^2

=

3 π

4 a^5 /^2

and

I=

1

2

×

3 π

4 a^5 /^2

.

Several other examples of integrals of multivalued functions around a variety

of contours are included in the exercises that follow.

24.14 Exercises

24.1 Find an analytic function ofz=x+iywhose imaginary part is

(ycosy+xsiny)expx.

24.2 Find a functionf(z), analytic in a suitable part of the Argand diagram, for which

Ref=

sin 2x

cosh 2y−cos 2x

.

Wherearethesingularitiesoff(z)?
24.3 Find the radii of convergence of the following Taylor series:

(a)

∑∞

n=2

zn

lnn

, (b)

∑∞

n=1

n!zn

nn

,

(c)

∑∞

n=1

znnlnn, (d)

∑∞

n=1

(

n+p

n

)n 2

zn,withpreal.

24.4 Find the Taylor series expansion about the origin of the functionf(z) defined by

f(z)=

∑∞

r=1

(−1)r+1sin

(pz

r

)

,

wherepis a constant. Hence verify thatf(z) is a convergent series for allz.
24.5 Determine the types of singularities (if any) possessed by the following functions
atz=0andz=∞:
(a) (z−2)−^1 , (b) (1 +z^3 )/z^2 , (c) sinh(1/z),
(d)ez/z^3 , (e)z^1 /^2 /(1 +z^2 )^1 /^2.

24.6 Identify the zeros, poles and essential singularities of the following functions:

(a) tanz, (b) [(z−2)/z^2 ] sin[1/(1−z)], (c) exp(1/z),

(d) tan(1/z), (e)z^2 /^3.