## 3

## Functions of Complex Variables

In this chapter, we deal with functions of complex variables which are useful in evaluating a large number of new definite integrals, the theory of differential equations, the study of electric fields, thermodynamics, and fluid mechanics.

##### 3.1 Basic Concepts

**Definition 3.1.** A *complex number z* is an ordered pair (*x*, *y*) of real numbers *x* and *y*. If *z* = (*x*, *y*) and *w* = (*u*, *v*) are two complex numbers, then their addition and multiplication are defined as

With these operations of addition and multiplication, the complex numbers satisfy the same arithmetic properties as do the real numbers.

If we write the real ...

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