Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.5 Multivalued functions and branch cuts

On the RHS let us writetas follows:

t=rexp[i(θ+2kπ)],

wherekis an integer. We then obtain

t^1 /n=exp

[

1

n

lnr+i

(θ+2kπ)

n

]

=r^1 /nexp

[

i

(θ+2kπ)

n

]

,

wherek=0, 1 ,...,n−1; for other values ofkwe simply recover the roots already found.
Thusthasndistinctnth roots.

24.5 Multivalued functions and branch cuts

In the definition of an analytic function, one of the conditions imposed was

that the function is single-valued. However, as shown in the previous section, the

logarithmic function, a complex power and a complex root are all multivalued.

Nevertheless, it happens that the properties of analytic functions can still be

applied to these and other multivalued functions of a complex variable provided

that suitable care is taken. This care amounts to identifying thebranch pointsof

the multivalued functionf(z) in question. Ifzis varied in such a way that its

path in the Argand diagram forms a closed curve that encloses a branch point,

then, in general,f(z) will not return to its original value.

For definiteness let us consider the multivalued functionf(z)=z^1 /^2 and express

zasz=rexpiθ. From figure 24.1(a), it is clear that, as the pointztraverses any

closed contourCthat does not enclose the origin,θwill return to its original

value after one complete circuit. However, for any closed contourC′that does

enclose the origin, after one circuitθ→θ+2π(see figure 24.1(b)). Thus, for the

functionf(z)=z^1 /^2 , after one circuit

r^1 /^2 exp(iθ/2) →r^1 /^2 exp[i(θ+2π)/2] =−r^1 /^2 exp(iθ/2).

In other words, the value off(z) changes around any closed loop enclosing the

origin; in this casef(z)→−f(z). Thusz= 0 is a branch point of the function

f(z)=z^1 /^2.

We note in this case that if any closed contour enclosing the origin is traversed

twicethenf(z)=z^1 /^2 returns to its original value. The number of loops around

a branch point required for any given functionf(z) to return to its original value

depends on the function in question, and for some functions (e.g. Lnz, which also

has a branch point at the origin) the original value is never recovered.

In order thatf(z) may be treated as single-valued, we may define abranch cut

in the Argand diagram. A branch cut is a line (or curve) in the complex plane

and may be regarded as an artificial barrier that we must not cross. Branch cuts

are positioned in such a way that we are prevented from making a complete