Difference between revisions of "Gudermannian"
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The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | ||
$$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$ | $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\sin(\mathrm{gd}(x))=\tanh(x),$$ | ||
| + | where $\sin$ denotes the [[sine]], $\mathrm{gd}$ denotes the [[Gudermannian]], and $\tanh$ denotes the [[tanh|hyperbolic tangent]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
<center>{{:*-integral functions footer}}</center> | <center>{{:*-integral functions footer}}</center> | ||
Revision as of 22:42, 25 August 2015
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$
Properties
Theorem: The following formula holds: $$\sin(\mathrm{gd}(x))=\tanh(x),$$ where $\sin$ denotes the sine, $\mathrm{gd}$ denotes the Gudermannian, and $\tanh$ denotes the hyperbolic tangent.
Proof: █
