14

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.

14.1 BASIC CONCEPTS

Definition 14.1.    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

 

z + w = (x, y) + (u, v) = (x + u, y + v)
zw = (x, y) (u, v) = (xu – yv, xv + yu).

 

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 number x as (x

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