18.2 Functions of a Complex Variable

Functions of (x, y) that depend only on the combination (x + iy) are called functions of a complex variable and functions of this kind that can be expanded in power series in this variable are of particular interest.

This combination (x + iy) is generally called z, and we can define such functions as zⁿ, exp(z), sin z, and all the standard functions of z as well as of x.

They are defined in exactly the same way the only difference being that they are actually complex valued functions, that is, they are vectors in this two dimensional complex number space, each with a real and an imaginary part (or component).

Most of the standard functions we have previously discussed have the property that their values are real when their arguments are real. The obvious exception is the square root function, which becomes imaginary for negative arguments.

Since we can multiply z by itself and by any other complex number, we can form any polynomial in z and any power series as well. We define the exponential and sine functions of z by their power series expansions which converge everywhere in the complex plane.

Since all the operations that produce standard functions can be applied to complex functions we can produce all the standard functions of a complex variable by the same steps as go to producing standard functions of real variables.