Difference between revisions of "Csch"
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| − | The hyperbolic cosecant function is defined by | + | __NOTOC__ |
| + | The hyperbolic cosecant function $\mathrm{csch} \colon \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \setminus \{0\}$ is defined by | ||
$$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ | $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ | ||
| − | where $\sinh$ denotes the [[Sinh|hyperbolic sine]]. | + | where $\sinh$ denotes the [[Sinh|hyperbolic sine]]. Since this function is [[one-to-one]], its [[inverse function]], the [[arccsch|inverse hyperbolic cosecant]] function is clear. |
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
File:Cschplot.png|Plot of $\mathrm{csch}$ on $[-5,5]$. | File:Cschplot.png|Plot of $\mathrm{csch}$ on $[-5,5]$. | ||
| − | File: | + | File:Complexcschplot.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{csch}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of hyperbolic cosecant]]<br /> | |
| − | + | [[Antiderivative of hyperbolic cosecant]]<br /> | |
| − | + | [[Relationship between csch and csc]]<br /> | |
| − | + | [[Relationship between cot, Gudermannian, and csch]]<br /> | |
| + | [[Relationship between csch, inverse Gudermannian, and cot]]<br /> | ||
| + | [[Pythagorean identity for coth and csch]]<br /> | ||
| − | + | =See Also= | |
| + | [[Arccsch]] | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Tanh|next=Sech}}: $4.5.4$ | ||
| + | |||
| + | {{:Hyperbolic trigonometric functions footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 23:35, 21 October 2017
The hyperbolic cosecant function $\mathrm{csch} \colon \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \setminus \{0\}$ is defined by $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ where $\sinh$ denotes the hyperbolic sine. Since this function is one-to-one, its inverse function, the inverse hyperbolic cosecant function is clear.
Plot of $\mathrm{csch}$ on $[-5,5]$.
Domain coloring of analytic continuation of $\mathrm{csch}$.
Properties
Derivative of hyperbolic cosecant
Antiderivative of hyperbolic cosecant
Relationship between csch and csc
Relationship between cot, Gudermannian, and csch
Relationship between csch, inverse Gudermannian, and cot
Pythagorean identity for coth and csch
See Also
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.5.4$
