Difference between revisions of "Sech"
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The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by | The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by | ||
| − | $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)} | + | $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)},$$ |
| − | Since this function is not [[one-to-one]], we define the [[arcsech|inverse hyperbolic secant function]] as the [[inverse function]] of $\mathrm{sech}$ restricted to $[0,\infty)$. | + | where $\cosh(z)$ denotes the [[cosh|hyperbolic cosine]]. Since this function is not [[one-to-one]], we define the [[arcsech|inverse hyperbolic secant function]] as the [[inverse function]] of $\mathrm{sech}$ restricted to $[0,\infty)$. |
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<gallery> | <gallery> | ||
Revision as of 22:58, 24 June 2016
The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by $$\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
See Also
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.5.5
