### 2.2  REVIEW OF COMPLEX VARIABLES, COMPLEX FUNCTIONS, AND THE s PLANE

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 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.

Figure 2.1 illustrates the mapping of a single-valued function from the s plane to the F(s) plane. Figure 2.2 illustrates the corresponding mapping for a multivalued function.

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.

#### A.  Analytic ...

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