special cases of hypergeometric function

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

F(a,\,b,\,c;\,x)\;=\;1+\frac{ab}{1!c}x+\frac{a(a+1)b(b+1)}{2!c(c+1)}x^{2}+% \frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}x^{3}+\ldots,

which are solutions of the hypergeometric equation

x(x-1)\frac{d^{2}y}{dx^{2}}+(c-(a+b+1))\frac{dy}{dx}-aby\;=\;0.

For example:

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$(1\!+\!x)^{n}\;=\;F(-n,\,1,\,1;\,-x)$
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$\ln(1\!+\!x)\;=\;xF(1,\,1,\,2;\,-x)$
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$\ln\frac{1+x}{1-x}\;=\;2xF(\frac{1}{2},\,1,\,\frac{3}{2};\,x^{2})$
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$\arcsin{x}\;=\;xF(\frac{1}{2},\,\frac{1}{2},\,\frac{3}{2};\,x^{2})$
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$\arctan{x}\;=\;xF(\frac{1}{2},\,1,\,\frac{3}{2};\,-x^{2})$
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$\sin(m\arcsin{x})\;=\;mxF(\frac{1+m}{2},\,\frac{1-m}{2},\,\frac{3}{2};\,x^{2})$
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$\cos(m\arcsin{x})\;=\;F(\frac{m}{2},\,-\frac{m}{2},\,\frac{1}{2};\,x^{2})$
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$T_{n}(x)\;=\;F(n,\,-n,\,\frac{1}{2};\,\frac{1-x}{2})$ (Chebyshev polynomials)
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$P_{n}(x)\;=\;F(-n,\,n+1,\,1;\,\frac{1-x}{2})$ (Legendre polynomials)
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$\displaystyle\int_{0}^{\frac{\pi}{2}}\!\frac{d\varphi}{\sqrt{1\!-\!x^{2}\sin^{% 2}\varphi}}$ $\;=\;\frac{\pi}{2}F(\frac{1}{2},\,\frac{1}{2},\,1;\,x^{2})$ (complete elliptic integral of 1st kind)
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$\displaystyle\int_{0}^{\frac{\pi}{2}}\!\sqrt{1\!-\!x^{2}\sin^{2}\varphi}\;d\varphi$ $\;=\;\frac{\pi}{2}F(-\frac{1}{2},\,\frac{1}{2},\,1;\,x^{2})$ (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