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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
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