Difference between revisions of "Csch"
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| − | [[File:Complex Csch.jpg| | + | The hyperbolic cosecant function is defined by |
| + | $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ | ||
| + | where $\sinh$ denotes the [[Sinh|hyperbolic sine]]. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Complex Csch.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{csch}$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | {{:Derivative of hyperbolic cosecant}} | ||
| + | |||
| + | <center>{{:Hyperbolic trigonometric functions footer}}</center> | ||
Revision as of 05:39, 20 March 2015
The hyperbolic cosecant function is defined by $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ where $\sinh$ denotes the hyperbolic sine.
- Complex Csch.jpg
Domain coloring of analytic continuation of $\mathrm{csch}$.
Contents
Properties
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z)=-\mathrm{csch}(z)\mathrm{coth}(z),$$ where $\mathrm{csch}$ denotes the hyperbolic cosecant and $\mathrm{coth}$ denotes the hyperbolic cotangent.
Proof
From the definition, $$\mathrm{csch}(z) = \dfrac{1}{\mathrm{sinh}(z)}.$$ Using the quotient rule, the derivative of sinh, and the definition of $\mathrm{coth}$, we compute $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z) &= \dfrac{0-\mathrm{cosh}(z)}{\mathrm{sinh}^2(z)} \\ &= -\mathrm{csch}(z)\mathrm{coth}(z), \end{array}$$ as was to be shown.
