Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.6 Singularities and zeros of complex functions

(a) (b) (c)

−i −i −i

i i i

y y y

x x x

r 1

r 2

θ 1

θ 2

z

Figure 24.2 (a) Coordinates used in the analysis of the branch points of

f(z)=(z^2 +1)^1 /^2 ; (b) one possible arrangement of branch cuts; (c) another

possible branch cut, which is finite.

(iii)z=−ibut notz=i,thenθ 1 →θ 1 ,θ 2 →θ 2 +2πand sof(z)→−f(z);

(iv) both branch points, thenθ 1 →θ 1 +2π,θ 2 →θ 2 +2πand sof(z)→f(z).

Thus, as expected,f(z) changes value around loops containing eitherz=iorz=−i(but
not both). We must therefore choose branch cuts that prevent us from making a complete
loop around either branch point; one suitable choice is shown in figure 24.2(b).
For thisf(z), however, we have noted that after traversing a loop containingbothbranch
points the function returns to its original value. Thus we may choose an alternative,finite,
branch cut that allows this possibility but still prevents us from making a complete loop
around just one of the points. A suitable cut is shown in figure 24.2(c).

24.6 Singularities and zeros of complex functions

A singular point of a complex functionf(z) is any point in the Argand diagram

at whichf(z) fails to be analytic. We have already met one sort of singularity,

the branch point, and in this section we will consider other types of singularity

as well as discuss the zeros of complex functions.

Iff(z) has a singular point atz=z 0 but is analytic at all points in some

neighbourhood containingz 0 but no other singularities, thenz=z 0 is called an

isolated singularity. (Clearly, branch points are not isolated singularities.)

The most important type of isolated singularity is thepole.Iff(z) has the form

f(z)=

g(z)

(z−z 0 )n

, (24.23)

wherenis a positive integer,g(z) is analytic at all points in some neighbourhood

containingz=z 0 andg(z 0 )=0,thenf(z) has apole of ordernatz=z 0 .An

alternative (though equivalent) definition is that

lim

z→z 0

[(z−z 0 )nf(z)]=a, (24.24)