Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

Then, by the principle of the argument, the number of zeros insideCis given by
the integer (2π)−^1 ∆C[argf(z)].
It can be shown that the zeros ofz^4 +z+ 1 lie one in each quadrant. Use the
above method to show that the zeros in the second and third quadrants have
|z|<1.
25.9 Prove that
∑∞

−∞

1

n^2 +^34 n+^18

=4π.

Carry out the summation numerically, say between−4 and 4, and note how
much of the sum comes from values near the poles of the contour integration.
25.10 This exercise illustrates a method of summing some infinite series.

(a) Determine the residues at all the poles of the function

f(z)=

πcotπz

a^2 +z^2

,

whereais a positive real constant.

(b) By evaluating, in two different ways, the integralIoff(z) along the straight

line joining−∞−ia/2and+∞−ia/2, show that

∑∞

n=1

1

a^2 +n^2

=

πcothπa

2 a

−

1

2 a^2

.

(c) Deduce the value of

∑∞

1 n

− (^2).
25.11 By considering the integral of
(
sinαz
αz

) 2

π

sinπz

,α<

π

2

,

around a circle of large radius, prove that

∑∞

m=1

(−1)m−^1

sin^2 mα

(mα)^2

=

1

2

.

25.12 Use the Bromwich inversion, and contours similar to that shown in figure 25.7(a),
to find the functions of which the following are the Laplace transforms:
(a) s(s^2 +b^2 )−^1 ;
(b)n!(s−a)−(n+1),withna positive integer ands>a;
(c) a(s^2 −a^2 )−^1 ,withs>|a|.
Compare your answers with those given in a table of standard Laplace transforms.
25.13 Find the functionf(t) whose Laplace transform is

f ̄(s)=e

−s−1+s

s^2

.

25.14 A functionf(t) has the Laplace transform

F(s)=

1

2 i

ln

(

s+i

s−i

)

,

the complex logarithm being defined by a finite branch cut running along the

imaginary axis from−itoi.

(a) Convince yourself that, fort>0,f(t) can be expressed as a closed contour

integral that encloses only the branch cut.