Difference between revisions of "Antiderivative of hyperbolic cosecant"
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<strong>[[Antiderivative of hyperbolic cosecant|Theorem]]:</strong> The following formula holds: | <strong>[[Antiderivative of hyperbolic cosecant|Theorem]]:</strong> The following formula holds: | ||
| − | $$\displaystyle\int \mathrm{csch}(z) | + | $$\displaystyle\int \mathrm{csch}(z)\mathrm{d}z = \log\left(\tanh\left(\frac{z}{2}\right)\right),$$ |
where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]], $\log$ denotes the [[logarithm]], and $\tanh$ denotes the [[tanh|hyperbolic tangent]]. | where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]], $\log$ denotes the [[logarithm]], and $\tanh$ denotes the [[tanh|hyperbolic tangent]]. | ||
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Revision as of 08:19, 16 May 2016
Theorem: The following formula holds: $$\displaystyle\int \mathrm{csch}(z)\mathrm{d}z = \log\left(\tanh\left(\frac{z}{2}\right)\right),$$ where $\mathrm{csch}$ denotes the hyperbolic cosecant, $\log$ denotes the logarithm, and $\tanh$ denotes the hyperbolic tangent.
Proof: █
