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Evaluate csch(x)
Evaluate cosh(x)
Evaluate sinh(x)
Evaluate arsech(x)
Evaluate exponential
Evaluate sec(x)
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Hyperbolic Secant Calculator

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

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

x =
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sech(x) =
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Definitions

General

The hyperbolic secant function is defined as:

\[ \mathrm{sech}\,{x} = \frac{1}{\cosh{x}} = \frac{2}{\mathrm{e}^x+\mathrm{e}^{-x}} \]

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

sech-graph

Series

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

\[ \begin{split} \mathrm{sech}\, x & = \sum_{n=0}^{\infty}\frac{ E_{2n} x^{2n}}{(2n)!} = \\ & = 1 - \frac{x^2}{2} + \frac{5x^4}{24} - \frac{61 x^6}{720} \cdots \end{split} \]

The above series converges for \( |x| < \frac{\pi}{2} \). En denotes the n-th Euler number .

Properties

The derivative of the hyperbolic secant function is:

\[ \left(\mathrm{sech}\,{x}\right)' = -\mathrm{sech}\,{x}\tanh{x} \]

The integral of the hyperbolic secant is given by:

\[ \int \mathrm{sech}\,{x} \, \mathrm{d}x =\arctan{\left( \sinh{x} \right)} +C \]

Identities

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

See also
Evaluate csch(x)
Evaluate cosh(x)
Evaluate sinh(x)
Evaluate arsech(x)
Evaluate exponential
Evaluate sec(x)
All evaluation tools