Difference between revisions of "Sech"

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The hyperbolic secant function is defined by
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$$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)}.$$
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The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by
 
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$$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)},$$
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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>
File:Arccos.png|Graph of $\mathrm{arccos}$ on $[-1,1]$.
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File:Sechplot.png|Graph of $\mathrm{sech}$ on $[-5,5]$.
File:Complex Sech.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sech}$.
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File:Complexsechplot.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sech}$.
 
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</gallery>
 
</div>
 
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=Properties=
 
=Properties=
{{:Derivative of sech}}
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[[Derivative of sech]]<br />
{{:Antiderivative of sech}}
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[[Antiderivative of sech]]<br />
{{:Relationship between cosine, Gudermannian, and sech}}
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[[Relationship between cosine, Gudermannian, and sech]]<br />
{{:Relationship between sech, inverse Gudermannian, and cos}}
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[[Relationship between sech, inverse Gudermannian, and cos]]<br />
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[[Pythagorean identity for tanh and sech]]<br />
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=See Also=
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[[Arcsech]]
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=References=
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Csch|next=Coth}}: $4.5.5$
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{{:Hyperbolic trigonometric functions footer}}
  
<center>{{:Hyperbolic trigonometric functions footer}}</center>
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[[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 $[0,\infty)$.

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

Arcsech

References

Hyperbolic trigonometric functions