Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS3.7 Hyperbolic functionsThe hyperbolic functions are the complex analogues of the trigonometric functions.The analogy may not be immediately apparent and their definitions may appearat first to be somewhat arbitrary. However, careful examination of their propertiesreveals the purpose of the definitions. For instance, their close relationship withthe trigonometric functions, both in their identities and in their calculus, meansthat many of the familiar properties of trigonometric functions can also be appliedto the hyperbolic functions. Further, hyperbolic functions occur regularly, and sogiving them special names is a notational convenience.3.7.1 DefinitionsThe two fundamental hyperbolic functions are cosh x and sinh x, which, as theirnames suggest, are the hyperbolic equivalents of cos x and sin x. They are definedby the following relations:cosh x = 1 2 (ex + e −x ), (3.38)sinh x = 1 2 (ex − e −x ). (3.39)Note that cosh x is an even function and sinh x is an odd function. By analogywith the trigonometric functions, the remaining hyperbolic functions aretanh x = sinh xcosh x = ex − e −xe x , (3.40)+ e−x sech x = 1cosh x = 2e x , (3.41)+ e−x cosech x = 1sinh x = 2e x ,− e−x (3.42)coth x = 1tanh x = ex + e −xe x .− e−x (3.43)All the hyperbolic functions above have been defined in terms of the real variablex. However, this was simply so that they may be plotted (see figures 3.11–3.13);the definitions are equally valid for any complex number z.3.7.2 Hyperbolic–trigonometric analogiesIn the previous subsections we have alluded to the analogy between trigonometricand hyperbolic functions. Here, we discuss the close relationship between the twogroups of functions.Recalling (3.32) and (3.33) we findcos ix = 1 2 (ex + e −x ),sin ix = 1 2 i(ex − e −x ).102