Advanced Engineering Mathematics, 7th Edition

17.7 Trigonometric and Hyperbolic Functions

INTRODUCTION

In this section we define the complex trigonometric and hyperbolic functions. Analogous to the complex functions e^z and Ln z defined in the previous section, these functions will agree with their real counterparts for real values of z. In addition, we will show that the complex trigonometric and hyperbolic functions have the same derivatives and satisfy many of the same identities as the real trigonometric and hyperbolic functions.

Trigonometric Functions

If x is a real variable, then Euler’s formula gives

e^ix = cos x + i sin x     and     e^−ix = cos x − i sin x.

By subtracting and then ...