5

**Complex numbers and hyperbolic functions**

This chapter is concerned with the representation and manipulation of complex numbers. Complex numbers pervade this book, underscoring their wide application in the mathematics of the physical sciences. Some elementary applications of complex numbers to the description of physical systems appear in later chapters, but only the basic tools are presented here.

5.1 The need for complex numbers

Although complex numbers occur in many branches of mathematics, they arise most directly out of solving polynomial equations. We examine a specific quadratic equation as an example.

Consider the quadratic equation

(5.1) |

Equation (5.1) has two solutions, *z*_{1} and *z*_{2}, such that

(5.2) |

Using the familiar ...

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