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Evaluate tanh(x)
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Evaluate cosh(x)
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Evaluate exponential
Evaluate cot(x)
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Hyperbolic cotangent calculator

- By Dr. Minas E. Lemonis, PhD - Updated: June 4, 2020
Home > Evaluation > Hyperbolic Cotangent

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

x =
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Result:
coth(x) =
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Table of Contents
-  Calculator
-  Definitions
-  General
-  Series
-  Properties
-  Identities
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Definitions

General

The hyperbolic cotangent function is defined as:

\coth{x} = \frac{\cosh{x}}{\sinh{x}} = \frac{\mathrm{e}^x+\mathrm{e}^{-x}}{\mathrm{e}^x-\mathrm{e}^{-x}} \quad, x\neq 0

The graph of the hyperbolic cotangent function is shown in the figure below. Unlike the trigonometric cotangent, the function is not periodic.

coth-graph

Series

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

\begin{split} \coth x & = x^{-1} + \sum_{n=1}^{\infty}\frac{ 2^{2n} B_{2n} x^{2n-1}}{(2n)!} = \\ & = x^{-1} + \frac{x}{3} - \frac{x^3}{45} + \frac{2 x^5}{945} \cdots \end{split}

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

Properties

The derivative of the hyperbolic cotangent function is:

\left(\coth{x}\right)' = 1-\coth^2{x} = -\frac{1}{\sinh^2{x}}

The integral of the hyperbolic cotangent is given by:

\int \coth{x} \, \mathrm{d}x = \ln\left(\sinh{x}\right) +C

Identities

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

See also
Evaluate tanh(x)
Evaluate sinh(x)
Evaluate cosh(x)
Evaluate arcoth(x)
Evaluate exponential
Evaluate cot(x)
All evaluation tools