1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 183

y

3

2

I '

_,,

4

• I
  -2


• 3

L 4

"' .....

--

x - I

• I

Figure 6.8 Vertical segments mapped ooto circular a.res by w = tao z.

How should we define the complex hyperbolic functions? We begin with

De finition 5.7: cosh z and sinh z

coshz= ~(e•+e-•) andsinhz= ~(e•-c•).

With these definitions in place, we can now easily create the other complex
hyperbolic trigonometric functions, provided the denominators in the following
expressions are not zero.

Definition 5.8: Complex hyperbolic functions

tanh z = ~:,';~:, coth z = ~~~! , sech z = CO:h z, and csch z = sinL.

As the series for the complex hyperbolic sine and cosine agree with the real
hyperbolic sine and cosine when z is real, the remaining complex hyperbolic
trigonometric functions likewise agree with their real counterparts. Many other
properties are also shared. We state several results without proof, as they follow
from the definit ions we gave using standard operations, such as the quotient rule
for derivatives. We ask you to establish some of these identities in the exercises.