# CHAPTER 13

# Complex Numbers and Functions. Complex Differentiation

The transition from “real calculus” to “complex calculus” starts with a discussion of *complex numbers* and their geometric representation in the *complex plane*. We then progress to *analytic functions* in Sec. 13.3. We desire functions to be analytic because these are the “useful functions” in the sense that they are differentiable in some domain and operations of complex analysis can be applied to them. The most important equations are therefore the Cauchy–Riemann equations in Sec. 13.4 because they allow a test of analyticity of such functions. Moreover, we show how the Cauchy–Riemann equations are related to the important *Laplace equation*.

The remaining sections of the chapter are devoted to elementary complex functions (exponential, trigonometric, hyperbolic, and logarithmic functions). These generalize the familiar real functions of calculus. Detailed knowledge of them is an absolute necessity in practical work, just as that of their real counterparts is in calculus.

*Prerequisite:* Elementary calculus.

*References and Answers to Problems:* App. 1 Part D, App. 2.

## 13.1 Complex Numbers and Their Geometric Representation

The material in this section will most likely be familiar to the student and serve as a review.

Equations without *real* solutions, such as *x*^{2} = −1 or *x*^{2} − 10*x* + 40 = 0, were observed early in history and ...