文档介绍：Functions of plex
Variable
Ultimately it was realized that to accept numbers that provided solutions to equations such as
x2 þ 1 ¼ 0 was no less meaningful than had been the extension of the real number system to admit a
solution for x þ 1 ¼ 0, or roots for x2 À 2 ¼ 0. plex number system was in place around 1700,
and by the early eenth century, mathematicians fortable with it. Physical theories took
on pleteness not possible without this foundation plex numbers and the analysis emanating
from it. The theorems of the differential and integral calculus plex 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 plex numbers which a variable z may assume there corresponds one or more
values of a variable w, then w is called a function of plex variable z, written w ¼ f ðzÞ. The
fundamental operations plex 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-, we can write w ¼ f ðzÞ¼uðx; yÞþivðx; yÞ, where u and v are
real functions of x and y.
EXAMPLE. w ¼ z2 ¼ðx þ iyÞ2 ¼ x2 À y2 þ 2ixy ¼ u þ iv so that uðx; yÞ¼x2 À y2; vðx; yÞ¼2xy. These are
called the real and imaginary parts of w ¼ z2 respectively.
plex 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 e2 ki ¼ 1, the general polar form of z is z ¼ eið 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 2kÞik¼ 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 plex variables) a single-valued function c