Difference between revisions of "Sech"
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| − | [ | + | The hyperbolic secant function is defined by |
| + | $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)}.$$ | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Arccos.png|Graph of $\mathrm{arccos}$ on $[-1,1]$. | ||
| + | File:Complex Sech.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sech}$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | {{:Derivative of sech}} | ||
| + | |||
| + | <center>{{:Hyperbolic trigonometric functions footer}}</center> | ||
Revision as of 05:44, 20 March 2015
The hyperbolic secant function is defined by $$\mathrm{sech}(z)=\dfrac{1}{\cosh(z)}.$$
Graph of $\mathrm{arccos}$ on $[-1,1]$.
- Complex Sech.jpg
Domain coloring of analytic continuation of $\mathrm{sech}$.
Contents
Properties
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z)=-\mathrm{sech}(z)\mathrm{tanh}(z),$$ where $\mathrm{sech}$ denotes the hyperbolic secant and $\mathrm{tanh}$ denotes the hyperbolic tangent.
Proof
From the definition, $$\mathrm{sech}(z) = \dfrac{1}{\mathrm{cosh}(z)}.$$ Using the quotient rule, the derivative of cosh, and the definition of $\mathrm{tanh}$, we see $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{sech}(z) &= \dfrac{0-\sinh(z)}{\cosh(z)^2} \\ &=-\mathrm{sech}(z)\mathrm{tanh}(z), \end{array}$$ as was to be shown.
