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NIST
4 Elementary FunctionsHyperbolic Functions

§4.28 Definitions and Periodicity

4.28.1 sinhz =ez-e-z2,
4.28.2 coshz =ez+e-z2,
4.28.3 coshz±sinhz =e±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(iz) =isinhz,
4.28.9 cos(iz) =coshz,
4.28.10 tan(iz) =itanhz,
4.28.11 csc(iz) =-icschz,
4.28.12 sec(iz) =sechz,
4.28.13 cot(iz) =-icothz.

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 2πi, and tanhz has period πi. The zeros of sinhz and coshz are z=ikπ and z=i(k+12)π, respectively, k.