Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.3 Location of zeros

φ=0

φ=0

Φ=0 Φ=0

y

x

π/α

z 0

w=zα

s

r

w 0

(a) (b) w^0 ∗

Figure 25.4 (a) An infinite conducting wedge with interior angleπ/αand a

line charge atz=z 0 ; (b) after the transformationw=zα, with an additional

imagechargeplacedatw=w∗ 0.

Substitutingw=zαinto the above shows that the required complex potential in the
originalz-plane is

f(z)=

q

2 π 0

ln

(

zα−z∗ 0 α

zα−zα 0

)

.

It should be noted that the appearance of a complex conjugate in the final

expression is not in conflict with the general requirement that the complex

potential be analytic. It isz∗that must not appear; here,z 0 ∗αis no more than a

parameter of the problem.

25.3 Location of zeros

The residue theorem, relating the value of a closed contour integral to the sum of

the residues at the poles enclosed by the contour, was discussed in the previous

chapter. One important practical use of an extension to the theorem is that of

locating the zeros of functions of a complex variable. The location of such zeros

has a particular application in electrical network and general oscillation theory,

since the complex zeros of certain functions (usually polynomials) give the system

parameters (usually frequencies) at which system instabilities occur. As the basis

of a method for locating these zeros we next prove three important theorems.

(i) Iff(z) has poles as its only singularities inside a closed contourCand is

not zero at any point onCthen
∮

C

f′(z)

f(z)

dz=2πi

∑

j

(Nj−Pj). (25.14)

HereNjis the order of thejth zero off(z) enclosed byC. SimilarlyPjis the

order of thejth pole off(z) insideC.

To prove this we note that, at each positionzj,f(z) can be written as

f(z)=(z−zj)mjφ(z), (25.15)