special cases of hypergeometric function

Many elementary (http://planetmath.org/ElementaryFunction) and non-elementary transcendental functionsMathworldPlanetmath may be expressed as special cases of the hypergeometric functionsDlmfDlmfDlmfMathworldPlanetmath

F(a,b,c;x)= 1+ab1!cx+a(a+1)b(b+1)2!c(c+1)x2+a(a+1)(a+2)b(b+1)(b+2)3!c(c+1)(c+2)x3+,

which are solutions of the hypergeometric equation

x(x-1)d2ydx2+(c-(a+b+1))dydx-aby= 0.

For example:

  • (1+x)n=F(-n, 1, 1;-x)

  • ln(1+x)=xF(1, 1, 2;-x)

  • ln1+x1-x= 2xF(12, 1,32;x2)

  • arcsinx=xF(12,12,32;x2)

  • arctanx=xF(12, 1,32;-x2)

  • sin(marcsinx)=mxF(1+m2,1-m2,32;x2)

  • cos(marcsinx)=F(m2,-m2,12;x2)

  • Tn(x)=F(n,-n,12;1-x2)  (Chebyshev polynomialsDlmfPlanetmath)

  • Pn(x)=F(-n,n+1, 1;1-x2)  (Legendre polynomialsMathworldPlanetmath)

  • 0π2dφ1-x2sin2φ =π2F(12,12, 1;x2)  (complete elliptic integralMathworldPlanetmath of 1st kind)

  • 0π21-x2sin2φ𝑑φ =π2F(-12,12, 1;x2)  (complete elliptic integral of 2nd kind)

Title special cases of hypergeometric function
Canonical name SpecialCasesOfHypergeometricFunction
Date of creation 2013-03-22 18:54:39
Last modified on 2013-03-22 18:54:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 33C05
Related topic FrobeniusMethod
Related topic IndexOfSpecialFunctions
Related topic GettingTaylorSeriesFromDifferentialEquation