Difference between revisions of "Taylor series for Gudermannian"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\dfrac{\mathrm{gd}(x)}{2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \mathrm{tanh}^{2k+1}(\frac{x}{2})}{2k+1},$$ | $$\dfrac{\mathrm{gd}(x)}{2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \mathrm{tanh}^{2k+1}(\frac{x}{2})}{2k+1},$$ | ||
where $\mathrm{gd}$ is the [[Gudermannian]] and $\tanh$ is the [[tanh|hyperbolic tangent]]. | where $\mathrm{gd}$ is the [[Gudermannian]] and $\tanh$ is the [[tanh|hyperbolic tangent]]. | ||
| − | + | ||
| − | + | ==Proof== | |
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| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 08:08, 8 June 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{gd}(x)}{2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \mathrm{tanh}^{2k+1}(\frac{x}{2})}{2k+1},$$ where $\mathrm{gd}$ is the Gudermannian and $\tanh$ is the hyperbolic tangent.
