### 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 *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*.

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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