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.