Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

Y

y

X

R

O x

Figure 25.5 A contour for locating the zeros of a polynomial that occur in

the first quadrant of the Argand diagram.

(iii)YO:z=iyand so argh=tan−^1 y/(y^4 + 1), which starts at O(R−^3 ) and finishes at

0asygoes from largeRto 0. It never reachesπ/2 becausey^4 +1=0 has no real

positive root. Thus ∆YO[argh]=0.

Hence for the complete contour ∆C[argh]=0+2π+0+O(R−^3 ), and, ifRis allowed
to tend to infinity, we deduce from (25.17) thath(z) has one zero in the first quadrant.
Furthermore, since the roots occur in conjugate pairs, a second root must lie in the fourth
quadrant, and the other pair must lie in the second and third quadrants.
To show that the zeros lie within the given annulus in thez-plane we apply Rouche’s ́
theorem, as follows.

(i) WithCas|z|=3/2,f=z^4 ,g=z+1. Now|f|=81/16 onCand|g|≤1+|z|<

5 / 2 < 81 /16. Thus, sincez^4 = 0 has four roots inside|z|=3/2, so also does

z^4 +z+1=0.

(ii) WithCas|z|=2/3,f=1,g=z^4 +z.Nowf=1onCand|g|≤|z^4 |+|z|=

16 /81+2/3=70/ 81 <1. Thus, sincef= 0 has no roots inside|z|=2/3, neither

does 1 +z+z^4 =0.

Hence the four zeros ofh(z)=z^4 +z+ 1 occur one in each quadrant and all lie between
the circles|z|=2/3and|z|=3/2.

A further technique useful for locating the zeros of functions is explained in

exercise 25.8.

25.4 Summation of series

We now turn to an application of contour integration which at first sight might

seem to lie in an unrelated area of mathematics, namely the summation of infinite

series. Sometimes a real infinite series with indexn, say, can be summed with the

help of a suitable complex function that has poles on the real axis at the various

positionsz=nwith the corresponding residues at those poles equal to the values

of the terms of the series. A worked example provides the best explanation of

how the technique is applied; other examples will be found in the exercises.