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Re: Simple integral over special functions---HOW?



Since the general results are known, the easiest method is to use
Gradshteyn & Ryzhik 7.132 ( note condition that 2 Re lambda >
Abs[Re[mu]] )and modify Integrate (see notebook below).

Bob Hanlon
____________________________

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    FormBox[
      RowBox[{
        RowBox[{
          RowBox[{\(\[Integral]\_\(-1\)\%1\), 
            RowBox[{
              RowBox[{\(\((1 - x_\^2)\)\^\[Lambda]_\), " ", 
                RowBox[{
                  SubsuperscriptBox[
                    TagBox["P",
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          \(\(\[Pi]\ 2\^\[Mu]\ \(\[CapitalGamma](\[Lambda] + \[Mu]\/2 +
1)\)\ 
                \(\[CapitalGamma](\[Lambda] - \[Mu]\/2 + 1)
                  \)\)\/\(\(\[CapitalGamma](\[Lambda] + \[Nu]\/2 +
3\/2)\)\ 
                \(\[CapitalGamma](\[Lambda] - \[Nu]\/2 + 1)\)\ 
                \(\[CapitalGamma](\(-\(\[Mu]\/2\)\) + \[Nu]\/2 + 1)\)\ 
                \(\[CapitalGamma](\(-\(\[Mu]\/2\)\) - \[Nu]\/2 + 1\/2)
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In a message dated 5/24/98 2:09:53 AM, morrison@phyast.nhn.ou.edu wrote:

>Hi. I have repeatedly run into trouble trying to get Mathematica to
>evaluate analytically simple integrals involving special functions. For
>example, the following integral has a simple analytic form:
>        Integrate[LegendreP[n,x]/Sqrt[1-x^2],{x,-1,+1}] When I enter the
>above into Mathematica, it returns the integral unevaluated unless I
>specify a value for n. Figuring that the problem was that Mathematica
>didn't realize that n is a non-negative integer, I did the following:
>        n/: IntegerQ[n] = True;
>        Integrate[LegendreP[n,x]/Sqrt[1-x^2], {x,-1,+1}, Assumptions ->
>n  >=  0]
>Again, Mathematica returned the integral unevaluated, along with the
>assumption.
>It's as though the information that n is an integer in the Global`
>context doesn't get communicated to the Integrate command. Can anyone
>tell me (a) whether the above method fully specifies n as an integer in
>all such situations (I have other failures where function definitions
>seem unaware of such a specification) and if not, how to properly
>specify a symbol as an integer and (b) how to make Integrate evaluate
>integrals such as the example above?



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