Re: Convert to hypergeometric function
- To: mathgroup at smc.vnet.net
- Subject: [mg62806] Re: Convert to hypergeometric function
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Mon, 5 Dec 2005 13:41:18 -0500 (EST)
- Organization: The University of Western Australia
- References: <dn0vfg$8k3$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <dn0vfg$8k3$1 at smc.vnet.net>, bd satish <bdsatish at gmail.com> wrote: > How do you express the following function functions in terms of > the hypergeometric function ? > > ExponentialIntegral Ei(x) Go to http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/ and click on the "Representations through more general functions" link. There you will find that a number of possible representations including ExpIntegralEi[z] == z HypergeometricPFQ[{1, 1}, {2, 2}, z] + (1/2) (Log[z] - Log[1/z]) + EulerGamma > LogarithmicIntegral Li(x) At http://functions.wolfram.com/06.36.26.0001.01 you will find that LogIntegral[z] == Log[z] HypergeometricPFQ[{1, 1}, {2, 2}, Log[z]] + (1/2) (Log[Log[z]] - Log[1/Log[z]]) + EulerGamma > Trigonometric functions: > sin(x) arcsin(x) > cos(x) arccos(x) > tan(x) arctan(x) > cot(x) arccot(x) > sec(x) arcsec(x) > cosec(x) arccosec(x) Similar answers for all these case. E.g., at http://functions.wolfram.com/01.10.26.0001.01 you will find that Csc[z] == 1/(z HypergeometricPFQ[{}, {3/2}, -(z^2/4)]) > Similarly the hyperbolic functions : sinh(x) cosh(x) tanh(x) ;;; > arcsinh(x) arccosh(x) arctanh(x) > > The logarithmic function: log(1+x) or log(x) > the gamma function : gamma(n) > the lambert function : lambertW(n,x) > > PLEASE provide the solutions for all the above cases (if possible) I expect that I don't need to go into details here ... Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul