Editing Hyperbolic functions (section)

==Taylor series expressions==
It is possible to express explicitly the [[Taylor series]] at zero (or the [[Laurent series]], if the function is not defined at zero) of the above functions.

<math display="block">\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>
This series is [[convergent series|convergent]] for every [[complex number|complex]] value of {{mvar|x}}. Since the function {{math|sinh ''x''}} is [[odd function|odd]], only odd exponents for {{math|''x''}} occur in its Taylor series.

<math display="block">\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>
This series is [[convergent series|convergent]] for every [[complex number|complex]] value of {{mvar|x}}. Since the function {{math|cosh ''x''}} is [[even function|even]], only even exponents for {{mvar|x}} occur in its Taylor series.

The sum of the sinh and cosh series is the [[infinite series]] expression of the [[exponential function]].

The following series are followed by a description of a subset of their [[domain of convergence]], where the series is convergent and its sum equals the function.
<math display="block">\begin{align}

               \tanh x &= x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \qquad \left |x \right | < \frac {\pi} {2} \\

               \coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \sum_{n=0}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, \qquad 0 < \left |x \right | < \pi \\

 \operatorname{sech} x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \qquad \left |x \right | < \frac {\pi} {2} \\

 \operatorname{csch} x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \sum_{n=0}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , \qquad 0 < \left |x \right | < \pi

\end{align}</math>

where:
*<math>B_n </math> is the ''n''th [[Bernoulli number]]
*<math>E_n </math> is the ''n''th [[Euler number]]