Difference between revisions of "Tanh"
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where $\mathrm{sinh}$ is the [[sinh|hyperbolic sine]] and $\mathrm{cosh}$ is the [[cosh|hyperbolic cosine]]. | where $\mathrm{sinh}$ is the [[sinh|hyperbolic sine]] and $\mathrm{cosh}$ is the [[cosh|hyperbolic cosine]]. | ||
| − | + | <div align="center"> | |
| + | <gallery> | ||
| + | File:Complex Tanh.jpg|[[Domain coloring]] of [[analytic continuation]] of $\tanh$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | {{:Derivative of tanh}} | ||
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| + | <center>{{:Hyperbolic trigonometric functions footer}}</center> | ||
Revision as of 05:34, 20 March 2015
The hyperbolic tangent is defined by the formula $$\mathrm{tanh}(z)=\dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)},$$ where $\mathrm{sinh}$ is the hyperbolic sine and $\mathrm{cosh}$ is the hyperbolic cosine.
- Complex Tanh.jpg
Domain coloring of analytic continuation of $\tanh$.
Contents
Properties
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z)=\mathrm{sech}^2(z),$$ where $\tanh$ denotes the hyperbolic tangent and $\mathrm{sech}$ denotes the hyperbolic secant.
Proof
From the definition, $$\tanh(z) = \dfrac{\sinh(z)}{\cosh(z)},$$ and so using the derivative of sinh, the derivative of cosh, the quotient rule, the Pythagorean identity for sinh and cosh, and the definition of the hyperbolic secant, $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z}\left[ \dfrac{\sinh(z)}{\cosh(z)} \right] \\ &= \dfrac{\cosh^2(z)-\sinh^2(z)}{\cosh^2(z)} \\ &= \dfrac{1}{\cosh^2(z)} \\ &= \mathrm{sech}^2(z), \end{array}$$ as was to be shown. █
