Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS4coth x2tanh x−2 −11 2x−2coth x−4Figure 3.13 Graphs of tanh x and coth x.metric functions transparent. The similarity in their calculus is discussed furtherin subsection 3.7.6.3.7.3 Identities of hyperbolic functionsThe analogies between trigonometric functions and hyperbolic functions havingbeen established, we should not be surprised that all the trigonometric identitiesalso hold for hyperbolic functions, with the following modification. Whereversin 2 x occurs it must be replaced by − sinh 2 x, and vice versa. Note that thisreplacement is necessary even if the sin 2 x is hidden, e.g. tan 2 x =sin 2 x/ cos 2 xand so must be replaced by (− sinh 2 x/ cosh 2 x)=− tanh 2 x.◮Find the hyperbolic identity analogous to cos 2 x +sin 2 x =1.Using the rules stated above cos 2 x is replaced by cosh 2 x,andsin 2 x by − sinh 2 x,andsothe identity becomescosh 2 x − sinh 2 x =1.This can be verified by direct substitution, using the definitions of cosh x and sinh x; see(3.38) and (3.39). ◭Some other identities that can be proved in a similar way aresech 2 x =1− tanh 2 x, (3.48)cosech 2 x =coth 2 x − 1, (3.49)sinh 2x =2sinhx cosh x, (3.50)cosh 2x =cosh 2 x + sinh 2 x. (3.51)104