Difference between revisions of "Gudermannian"
From specialfunctionswiki
| Line 3: | Line 3: | ||
=Properties= | =Properties= | ||
| − | + | {{:Relationship between sine, Gudermannian, and tanh}} | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
<center>{{:*-integral functions footer}}</center> | <center>{{:*-integral functions footer}}</center> | ||
Revision as of 22:43, 25 August 2015
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$
Contents
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.
