Complex Functions

Complex Functions
Definition
Complex analysis is concerned with complex functions that are differentiable in some domain. Hence we should first say what we mean by a complex function. The concepts of limit and derivative in complex will be similar to that in calculus. Nevertheless it needs great attention because it will show interesting basic differences between real and complex calculus. Recall from calculus that a real function f defined on a set S of real numbers (usually an interval) is a rule that assigns to every x in S a real number f(x), called the value of f at x. Now in complex, S is a set of complex numbers. And a function f defined on S is a rule that assigns to every z in S a complex number w, called the value of f at z.
w=f(z)

Here z varies in S and is called a complex variable. The set S is called the domain of definition of f or, briefly, the domain of f. (In most cases S will be open and connected, thus a domain as defined just before).

Example: f(z)=z^2+3z+1 is a complex function defined for all z; that is, its domain S is the whole complex plane. The set of all values of a function f is called the range of f. w is complex, and it can be written as:

w=f(z)=u(x,y)+i v(x,y)