Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

y y y

x x x

C

θ θ

r r

(a) (b) (c)

C′

Figure 24.1 (a) A closed contour not enclosing the origin; (b) a closed

contour enclosing the origin; (c) a possible branch cut forf(z)=z^1 /^2.

circuit around any one branch point, and so the function in question remains

single-valued.

For the functionf(z)=z^1 /^2 , we may take as a branch cut any curve starting

at the originz= 0 and extending out to|z|=∞in any direction, since all

such curves would equally well prevent us from making a closed loop around

the branch point at the origin. It is usual, however, to take the cut along either

the real or the imaginary axis. For example, in figure 24.1(c), we take the cut as

the positive real axis. By agreeing not to cross this cut, we restrictθto lie in the

range 0≤θ< 2 π, and so keepf(z) single-valued.

These ideas are easily extended to functions with more than one branch point.

Find the branch points off(z)=

√

z^2 +1, and hence sketch suitable arrangements of

branch cuts.

We begin by writingf(z)as

f(z)=

√

z^2 +1=

√

(z−i)(z+i).

As shown above, the functiong(z)=z^1 /^2 has a branch point atz= 0. Thus we might
expectf(z) to have branch points at values ofzthat make the expression under the square
root equal to zero, i.e. atz=iandz=−i.
As shown in figure 24.2(a), we use the notation
z−i=r 1 expiθ 1 and z+i=r 2 expiθ 2.

We can therefore writef(z)as

f(z)=

√

r 1 r 2 exp(iθ 1 /2) exp(iθ 2 /2) =

√

r 1 r 2 exp

[

i(θ 1 +θ 2 )/ 2

]

.

Let us now consider howf(z) changes as we make one complete circuit around various
closed loopsCin the Argand diagram. IfCencloses

(i) neither branch point, thenθ 1 →θ 1 ,θ 2 →θ 2 and sof(z)→f(z);

(ii)z=ibut notz=−i,thenθ 1 →θ 1 +2π,θ 2 →θ 2 and sof(z)→−f(z);