Difference between revisions of "Sech"
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| − | The hyperbolic secant function is defined by | + | __NOTOC__ |
| − | $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)}.$$ | + | The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by |
| − | + | $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)},$$ | |
| + | where $\cosh(z)$ denotes the [[cosh|hyperbolic cosine]]. Since this function is not [[one-to-one]], we define the [[arcsech|inverse hyperbolic secant function]] as the [[inverse function]] of $\mathrm{sech}$ restricted to $[0,\infty)$. | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
File:Sechplot.png|Graph of $\mathrm{sech}$ on $[-5,5]$. | File:Sechplot.png|Graph of $\mathrm{sech}$ on $[-5,5]$. | ||
| − | File: | + | File:Complexsechplot.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sech}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of sech]]<br /> | |
| − | + | [[Antiderivative of sech]]<br /> | |
| − | + | [[Relationship between cosine, Gudermannian, and sech]]<br /> | |
| − | + | [[Relationship between sech, inverse Gudermannian, and cos]]<br /> | |
| + | [[Pythagorean identity for tanh and sech]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[Arcsech]] | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Csch|next=Coth}}: $4.5.5$ | ||
| + | |||
| + | {{:Hyperbolic trigonometric functions footer}} | ||
| − | + | [[Category:SpecialFunction]] | |
Latest revision as of 23:35, 21 October 2017
The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)},$$ where $\cosh(z)$ denotes the hyperbolic cosine. Since this function is not one-to-one, we define the inverse hyperbolic secant function as the inverse function of $\mathrm{sech}$ restricted to $[0,\infty)$.
Graph of $\mathrm{sech}$ on $[-5,5]$.
Domain coloring of analytic continuation of $\mathrm{sech}$.
Properties
Derivative of sech
Antiderivative of sech
Relationship between cosine, Gudermannian, and sech
Relationship between sech, inverse Gudermannian, and cos
Pythagorean identity for tanh and sech
See Also
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.5.5$
