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

 

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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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