University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 2. Complex Functions

2.1  Single-Valued and Multiple-Valued Functions

a.  Let be a set of complex numbers. If for every there corresponds one or several complex numbers, following a given rule, then the function is said to be defined on and we write .

is called the function's domain of definition.

b.  If to each , there corresponds one and only one value , function is said to be a single-valued function. If at least one has several corresponding values, then the function is called a multi-valued function.

c.  If , and we separate the real component from the imaginary component of , we get

When , the real component of , the imaginary component of are real functions.

d.  Examples:

1.  Function is single-valued, defined on the full z-plane, except .

If , then

2.  Polynomial of complex variable with a complex coefficient is a single-valued function defined on the -plane.

3.  Rational function is a single-valued function defined on the full -plane, except the points vanishing the denominator.

4.  Function is defined on the full -plane, and is three-valued function, since cube root has 3 values in a complex plane, for example:

, ,

5.  A point that, when we move about it, has the initial value of a multi-valued function, and is transforming into another of its values, is called a branch point.

2.2  Limits of Complex Functions

a.  Complex number is the limit of function , when , if for every there exists such that every holding , there holds .

It is denoted .

b.   if, for every , there exists , such that for every , then there holds .

c.   if, for every , there exists a , such that for every holding , , then there holds .

d.  Let and , . The limit exists if, and only if, the limits

exist.

2.3  Continuity

a.  Function is continuous at point if it is defined on the neighborhood of this point and .

b.  Function is continuous in a domain if it is continuous at every point of it.

c.  If functions and are continuous on , then also , , when , are continuous on .

d.  Function is continuous at point if and only if real functions , are continuous at .

e.  Function is uniformly continuous on , if for every there exists such that for every on holding there holds .

f.  Cantor’s theorem: A function continuous on bounded and closed domain is uniformly continuous on it.