Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.7 HYPERBOLIC FUNCTIONS

sechx

coshx

x

1

2

3

4

− 2 − 1 1 2

Figure 3.11 Graphs of coshxand sechx.

cosechx

cosechx

sinhx

x

2

4

− 2

− 4

− 2 − 1 1 2

Figure 3.12 Graphs of sinhxand cosechx.

Hence, by the definitions given in the previous subsection,

coshx=cosix, (3.44)

isinhx=sinix, (3.45)

cosx=coshix, (3.46)

isinx= sinhix. (3.47)

These useful equations make the relationship between hyperbolic and trigono-