Digital Library of Mathematical Functions
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4 Elementary FunctionsHyperbolic Functions

§4.28 Definitions and Periodicity

4.28.1 sinhz =z--z2,
4.28.2 coshz =z+-z2,
4.28.3 coshz±sinhz =±z,
4.28.4 tanhz =sinhzcoshz,
4.28.5 cschz =1sinhz,
4.28.6 sechz =1coshz,
4.28.7 cothz =1tanhz.

Relations to Trigonometric Functions

4.28.8 sin(z) =sinhz,
4.28.9 cos(z) =coshz,
4.28.10 tan(z) =tanhz,
4.28.11 csc(z) =-cschz,
4.28.12 sec(z) =sechz,
4.28.13 cot(z) =-cothz.

As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.

Periodicity and Zeros

The functions sinhz and coshz have period 2π, and tanhz has period π. The zeros of sinhz and coshz are z=kπ and z=(k+12)π, respectively, k.