Difference between revisions of "Antiderivative of hyperbolic cosecant"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
| − | $$\displaystyle\int \mathrm{csch}(z)\mathrm{d}z = \log\left(\tanh\left(\frac{z}{2}\right)\right),$$ | + | $$\displaystyle\int \mathrm{csch}(z)\mathrm{d}z = \log\left(\tanh\left(\frac{z}{2}\right)\right)+C,$$ |
where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]], $\log$ denotes the [[logarithm]], and $\tanh$ denotes the [[tanh|hyperbolic tangent]]. | where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]], $\log$ denotes the [[logarithm]], and $\tanh$ denotes the [[tanh|hyperbolic tangent]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:55, 8 June 2016
Theorem
The following formula holds: $$\displaystyle\int \mathrm{csch}(z)\mathrm{d}z = \log\left(\tanh\left(\frac{z}{2}\right)\right)+C,$$ where $\mathrm{csch}$ denotes the hyperbolic cosecant, $\log$ denotes the logarithm, and $\tanh$ denotes the hyperbolic tangent.
