Taylor series of hyperbolic functions

The differentiation rules


of the hyperbolic functionsDlmfMathworldPlanetmath imply

d2ndx2ncoshx=coshx,d2n+1dx2n+1coshx=sinhx  (n=0, 1, 2,).

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


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

coshx= 1+x22!+x44!+=n=0x2n(2n)!. (1)

Similarly, one can derive for the hyperbolic sine the expansion

sinhx=x+x33!+x55!+=n=0x2n+1(2n+1)!. (2)

Both series converge (http://planetmath.org/AbsoluteConvergence) and the functionsMathworldPlanetmath 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


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

tanhx=x-13x3+215x5-17315x7+-  (|x|<π2). (3)

The coefficients of this power seriesMathworldPlanetmath may be expressed with the Bernoulli numbersDlmfDlmfMathworldPlanetmathPlanetmath.

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