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tanh(z)

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3-d plot

Hyperbolic functions:

The hyperbolic tangent tanh(z) is (like in the real case) given by the quotient of sinh and cosh:
tanh(z) = sinh(z)/cosh(z) = ( e^z- e^-z )/( e^z+ e^-z ).
This function is periodic in the direction of the imaginary axis (it has period i Pi).