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MathGroup Archive 2006

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Re: Implicit integration of finite alternating series of hypergeometric (2F1) functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65424] Re: Implicit integration of finite alternating series of hypergeometric (2F1) functions
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 31 Mar 2006 06:09:19 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <e0gcrs$i6k$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <e0gcrs$i6k$1 at smc.vnet.net>,
 "Mark Smith" <dsummoner at hotmail.com> wrote:

> I am having a problem with Mathematica in determining a closed form
> analytical solution for the implicit integral of the following: 

What do you mean by an implicit integral?

> -(a/Pi)*Cos[Pi*(t-b)/a]*Hypergeometric2F1[0.5,0.5*(1-n),1.5,(Cos[Pi*(t-b)/a])^
> 2]*c + d

Note that, as far as Mathematica is concerned, the floating point number 
0.5 is _not_ the same as the exact rational number 1/2. If you want to 
compute an integral _exactly_ you should use _exact_ input.

> In this equation the terms a, b, c and d are fixed constants for the
> problem.  The term n is also a constant with value greater than zero.
> The term t is is the variable. 

So I assume that you computing an indefinite integral with respect to t?

Note that 

> Mathematica returns the input line, as an output line, without an evaluation.

Which means that it _cannot_ compute this integral (not directly, 
anyway). It can compute the indefinite integral of 

  Integrate[Cos[Pi (t-b)/a]^(2m+1), t]

which appears in the m-th term of the Hypergeometric2F1 function -- but 
this integral is another Hypergeometric2F1 function.

> When I specify n, a priori, with respect to the integration operation,
> Mathematica has no problem with performing the integration.  

This is usually the case.

> I would, however, like a closed form analytical solution or a family of
> solutions without the a priori specification of n. 

This is, generally, a much harder problem. Your integrand can be 
expressed as a Beta function (using FunctionExpand) but, because of the 
complexity of this expression, I would be surpised if closed-form 
integrals for general n can be obtained.

A change of variables, y == Cos[Pi (t-b)/a], formally leads to (part of) 
the integral being expressed as a MeijerG function: 

  Integrate[y Hypergeometric2F1[1/2, 1/2 - n/2, 3/2, y^2]/Sqrt[1-y^2], y]

but I'm not sure if that will be useful for you.

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


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