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Evaluate sech(x)
Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arcsch(x)
Evaluate exponential
Evaluate csc(x)
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Hyperbolic Cosecant Calculator

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

This tool evaluates the hyperbolic cosecant of a number: csch(x). Enter the argument x below.

x =
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csch(x) =
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Definitions

General

The hyperbolic cosecant function is defined as:

\[ \mathrm{csch}\,{x} = \frac{1}{\sinh{x}} = \frac{2}{\mathrm{e}^x-\mathrm{e}^{-x}} \]

The graph of the hyperbolic cosecant function is shown in the figure below.

csch-graph

Series

All hyperbolic functions can be defined in an infinite series form. Hyperbolic cosecant function can be written as:

\[ \begin{split} \mathrm{csch}\, x & = \frac{1}{x} - \sum_{n=1}^{\infty}\frac{\left(2^{2n}-2\right) B_{2n} x^{2n-1}}{(2n)!} = \\ & = \frac{1}{x} - \frac{x}{6} + \frac{7x^3}{360} - \frac{31 x^5}{15120} \cdots \end{split} \]

The above series converges for \( 0 < |x| < \pi \). Bn denotes the n-th Bernulli number.

Properties

The derivative of the hyperbolic cosecant function is:

\[ \left(\mathrm{csch}\,{x}\right)' = -\mathrm{csch}\,{x}\coth{x} \quad, x\neq 0 \]

The integral of the hyperbolic cosecant is given by:

\[ \int \mathrm{csch}\,{x} \, \mathrm{d}x =\ln{\left( \tanh{\frac{x}{2}} \right)} +C \]

Identities

\[ \begin{split} & \mathrm{csch}\,{\left(-x\right)} & = -\mathrm{csch}\,{x} \\ \\ & \mathrm{csch}\,{\left(2 x\right)} & = \frac{1}{2} \mathrm{csch}\,{x} \, \mathrm{sech}\,{x} \\ \\ & \mathrm{csch}\,{\left(x + y\right)} & = \frac{1}{\cosh{x}\sinh{y} + \sinh{x}\cosh{y}} \\ \\ & \mathrm{csch}\,{\left(\frac{x}{2}\right)} & = \sqrt{\frac{2}{\cosh{x}-1}} \\ \\ & \mathrm{csch}^2\, {x} & = \coth^2{x}-1 \\ \\ \end{split} \]

See also
Evaluate sech(x)
Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arcsch(x)
Evaluate exponential
Evaluate csc(x)
All evaluation tools