# Taylor series of hyperbolic functions

The differentiation rules

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

of the hyperbolic functions imply

 $\frac{d^{2n}}{dx^{2n}}\cosh{x}\;=\;\cosh{x},\quad\frac{d^{2n+1}}{dx^{2n+1}}% \cosh{x}\;=\;\sinh{x}\qquad(n=0,\,1,\,2,\,\ldots).$

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

 $f(x)\;=\;f(0)+\frac{f^{\prime}(0)}{1!}x+\frac{f^{\prime\prime}(0)}{2!}x^{2}+% \frac{f^{\prime\prime\prime}(0)}{3!}x^{3}+\ldots$

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

 $\displaystyle\cosh{x}\;=\;1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\ldots\;=\;\sum_% {n=0}^{\infty}\frac{x^{2n}}{(2n)!}.$ (1)

Similarly, one can derive for the hyperbolic sine the expansion

 $\displaystyle\sinh{x}\;=\;x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\ldots\;=\;\sum_% {n=0}^{\infty}\frac{x^{2n+1}}{(2n\!+\!1)!}.$ (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

 $\cosh{x}\;=\;\cos{ix},\qquad\sinh{x}\;=\;-i\sin{ix}.$

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

 $\displaystyle\tanh{x}\;=\;x-\frac{1}{3}x^{3}+\frac{2}{15}x^{5}-\frac{17}{315}x% ^{7}+-\ldots\qquad(|x|<\frac{\pi}{2}).$ (3)

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

Title Taylor series of hyperbolic functions TaylorSeriesOfHyperbolicFunctions 2013-03-22 19:07:04 2013-03-22 19:07:04 pahio (2872) pahio (2872) 7 pahio (2872) Derivation msc 30B10 msc 26A09 HyperbolicIdentities HigherOrderDerivatives