Hyperbolic functions: Difference between revisions

Revision as of 19:57, 23 July 2005

Revision as of 20:09, 23 July 2005

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:<math>\operatorname{csch}(x) = \frac{1}{\sinh(x)} = \frac {2} {e^x - e^{-x}} = \imath \csc(\imath x)</math>

::(''hyperbolic cosecant'', pronounced "cosheck" or "cosech")

==Series definition==

It is possible to express the above functions as series:

:<math>\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} +\cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}</math>

:<math>\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}</math>

:<math>\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = </math>

:<math>\coth x = </math>

:<math>\sech x = </math>

:<math>\csch x = </math>

where

:<math>B_n \,</math> is the nth [[Bernoulli number]]

:<math>E_n \,</math> is the nth [[Euler number]]

==Relationship to regular trigonometric functions==