# Csch

From specialfunctionswiki

The hyperbolic cosecant function $\mathrm{csch} \colon \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \setminus \{0\}$ is defined by $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ where $\sinh$ denotes the hyperbolic sine. Since this function is one-to-one, its inverse function, the inverse hyperbolic cosecant function is clear.

Domain coloring of analytic continuation of $\mathrm{csch}$.

# Properties

Derivative of hyperbolic cosecant

Antiderivative of hyperbolic cosecant

Relationship between csch and csc

Relationship between cot, Gudermannian, and csch

Relationship between csch, inverse Gudermannian, and cot

# See Also

# References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.5.4$

**Hyperbolic trigonometric functions**