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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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