Taylor series of hyperbolic functions

The differentiation rules

$$\frac{d}{dx}\cosh x=\sinh x,\frac{d}{dx}\sinh x=\cosh x$$

of the hyperbolic functions imply

$$\frac{d^{2n}}{d{x}^{2n}}\cosh x=\cosh x,\frac{d^{2n+1}}{d{x}^{2n+1}}\cosh x=\sinh x\mathit{\hspace{1em}\hspace{1em}}(n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }).$$

In the origin  $x=0$,  all even (http://planetmath.org/Even)-order derivatives of the hyperbolic cosine have the value 1, but the odd (http://planetmath.org/Odd)-order derivatives vanish.  Thus the Taylor series expansion

$$f(x)=f(0)+\frac{f^{\prime }(0)}{1!}x+\frac{f^{\prime \prime }(0)}{2!}{x}^2+\frac{f^{\prime \prime \prime }(0)}{3!}{x}^3+\mathrm{\dots }$$

of  $f(x):=\cosh x$  contains only the terms of even degree and writes simply

$\cosh x=\mathrm{\hspace{0.33em}1}+{\displaystyle \frac{x^2}{2!}}+{\displaystyle \frac{x^4}{4!}}+\mathrm{\dots }={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{x^{2n}}{(2n)!}}.$ (1)

Similarly, one can derive for the hyperbolic sine the expansion

$\sinh x=x+{\displaystyle \frac{x^3}{3!}}+{\displaystyle \frac{x^5}{5!}}+\mathrm{\dots }={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{x^{2n+1}}{(2n+1)!}}.$ (2)

Both series converge (http://planetmath.org/AbsoluteConvergence) and the functions for all real (and complex) values of $x$.  Comparing the expansions (1) and (2) with the corresponding ones of the circular functions cosine and sine, one sees easily that

$$\cosh x=\cos ix,\sinh x=-i\sin ix.$$

As for the Taylor expansion of the third important hyperbolic function tangens hyperbolica (http://planetmath.org/HyperbolicFunctions) tanh, it is obtained via division of the Taylor series (http://planetmath.org/TaylorSeriesViaDivision) (2) and (1); the begin of the quotient series is

(3)

The coefficients of this power series may be expressed with the Bernoulli numbers.

Title	Taylor series of hyperbolic functions
Canonical name	TaylorSeriesOfHyperbolicFunctions
Date of creation	2013-03-22 19:07:04
Last modified on	2013-03-22 19:07:04
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Derivation
Classification	msc 30B10
Classification	msc 26A09
Related topic	HyperbolicIdentities
Related topic	HigherOrderDerivatives