Page 401 - Schaum's Outline of Theory and Problems of Advanced Calculus

Functions of a Complex

Variable

Ultimately it was realized that to accept numbers that provided solutions to equations such as
2
x þ 1 ¼ 0 was no less meaningful than had been the extension of the real number system to admit a
2
solution for x þ 1 ¼ 0, or roots for x 2 ¼ 0. The complex number system was in place around 1700,
and by the early nineteenth century, mathematicians were comfortable with it. Physical theories took
on a completeness not possible without this foundation of complex numbers and the analysis emanating
from it. The theorems of the differential and integral calculus of complex functions introduce math-
ematical surprises as well as analytic refinement. This chapter is a summary of the basic ideas.

FUNCTIONS

If to each of a set of complex numbers which a variable z may assume there corresponds one or more
values of a variable w, then w is called a function of the complex variable z, written w ¼ f ðzÞ. The
fundamental operations with complex numbers have already been considered in Chapter 1.
A function is single-valued if for each value of z there corresponds only one value of w; otherwise it is
multiple-valued or many-valued.In general, we can write w ¼ f ðzÞ¼ uðx; yÞþ ivðx; yÞ, where u and v are
real functions of x and y.

2
2
2
2
2
2
EXAMPLE. w ¼ z ¼ðx þ iyÞ ¼ x y þ 2ixy ¼ u þ iv so that uðx; yÞ¼ x y ; vðx; yÞ¼ 2xy. These are
2
called the real and imaginary parts of w ¼ z respectively.
In complex variables, multiple-valued functions often are replaced by a specially constructed single-
valued function with branches. This idea is discussed in a later paragraph.
EXAMPLE. Since e 2 ki ¼ 1, the general polar form of z is z ¼ e ið þ2 kÞ . This form and the fact that the logarithm
and exponential functions are inverse leads to the following definition of ln z
ln z ¼ ln þð þ 2 kÞi k ¼ 0; 1; 2; ... ; n .. .
Each value of k determines a single-valued function from this collection of multiple-valued functions. These
are the branches from which (in the realm of complex variables) a single-valued function can be constructed.
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