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