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Elementary Functions > Tanh[z] > Introduction to the hyperbolic functions

The best-known properties and formulas for hyperbolic functions

Real values for real arguments

For real values of argument , the values of all the hyperbolic functions are real (or infinity).

In the points , the values of the hyperbolic functions are algebraic. In several cases, they can even be rational numbers, , or (e.g. , , or ). They can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

Simple values at zero

All hyperbolic functions has rather simple values for arguments and :


All hyperbolic functions are defined for all complex values of , and they are analytical functions of over the whole complex ‐plane and do not have branch cuts or branch points. The two functions and are entire functions with an essential singular point at . All other hyperbolic functions are meromorphic functions with simple poles at points (for and ) and at points (for and ).


All hyperbolic functions are periodic functions with a real period ( or ):

Parity and symmetry

All hyperbolic functions have parity (either odd or even) and mirror symmetry:

Simple representations of derivatives

The derivatives of all hyperbolic functions have simple representations that can be expressed through other hyperbolic functions:

Simple differential equations

The solutions of the simplest second‐order linear ordinary differential equation with constant coefficients can be represented through and . The other hyperbolic functions satisfy first‐order nonlinear differential equations:

All six hyperbolic functions satisfy first‐order nonlinear differential equations: