The design of control systems depends greatly on the application of complex-variable theory. In what follows, the complex variable s is composed of a real part σ and an imaginary part ω, where
In the complex s plane, σ is plotted horizontally and jω vertically. A complex function F(s) is considered to be a function of the complex variable s if there is at least one value of F(s) for every value of s. The function F(s) will have real and imaginary components, because s has real and imaginary components, and it has the following form:
If there is only one value of F(s) for every value of s, the function F(s) is called a single-valued function. However, if there is more than one point in the F(s) plane for every value of s, then F(s) is a multivalued function. Most complex functions used in linear control systems are single-valued functions of s.
Five notions of complex-variable theory that are important to the control-systems engineer are those of analytic functions, ordinary points, singularities, poles, and zeros of a function.