Advanced Engineering Mathematics, 7th Edition

17.6 Exponential and Logarithmic Functions

INTRODUCTION

In this and the next section, we shall examine the exponential, logarithmic, trigonometric, and hyperbolic functions of a complex variable z. Although the definitions of these complex functions are motivated by their real variable analogues, the properties of these complex functions will yield some surprises.

Exponential Function

Recall that in real variables the exponential function f(x) = e^x has the properties

f′(x) = f(x)     and     f(x₁ + x₂) = f(x₁)f(x₂). (1)

We certainly want the definition of the complex function f(z) = e^z, where z = x + iy, to reduce e^x for y = 0 and to possess ...