Showing content from https://dlmf.nist.gov/4.28 below:
§4.28 Definitions and Periodicity ⣠Hyperbolic Functions ⣠Chapter 4 Elementary Functions
§4.28 Definitions and Periodicity 4.28.1 sinh â¡ z = ez â e â z 2 , 4.28.2 cosh â¡ z = ez + e â z 2 , 4.28.3 cosh â¡ z ± sinh â¡ z = e ± z , 4.28.4 tanh â¡ z = sinh â¡ z cosh â¡ z , 4.28.5 csch â¡ z = 1 sinh â¡ z , 4.28.6 sech â¡ z = 1 cosh â¡ z , 4.28.7 coth â¡ z = 1 tanh â¡ z . Relations to Trigonometric Functions 4.28.8 sin â¡ ( i ⢠z ) = i ⢠sinh â¡ z , 4.28.9 cos â¡ ( i ⢠z ) = cosh â¡ z , 4.28.10 tan â¡ ( i ⢠z ) = i ⢠tanh â¡ z , 4.28.11 csc â¡ ( i ⢠z ) = â i ⢠csch â¡ z , 4.28.12 sec â¡ ( i ⢠z ) = sech â¡ z , 4.28.13 cot â¡ ( i ⢠z ) = â i ⢠coth â¡ z .
As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.
Periodicity and Zeros
The functions sinh â¡ z and cosh â¡ z have period 2 â¢ Ï â¢ i , and tanh â¡ z has period Ï â¢ i . The zeros of sinh â¡ z and cosh â¡ z are z = i ⢠k â¢ Ï and z = i ⢠( k + 1 2 ) â¢ Ï , respectively, k â ⤠.
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