# Tanh

From specialfunctionswiki

The hyperbolic tangent is defined by the formula $$\mathrm{tanh}(z)=\dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)},$$ where $\mathrm{sinh}$ is the hyperbolic sine and $\mathrm{cosh}$ is the hyperbolic cosine.

Domain coloring of $\tanh$.

# Properties[edit]

Derivative of tanh

Antiderivative of tanh

Relationship between tanh and tan

Relationship between tan and tanh

Relationship between sine, Gudermannian, and tanh

Relationship between tanh, inverse Gudermannian, and sin

Taylor series for Gudermannian

Pythagorean identity for tanh and sech

Period of tanh

Tanh is odd

Tanh of a sum

Halving identity for tangent (1)

Halving identity for tangent (2)

Halving identity for tangent (3)

Doubling identity for sinh (2)

# See Also[edit]

# References[edit]

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.5.3$

**Hyperbolic trigonometric functions**