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I am trying to generate a function that represents essentially a Beta Distribution that revolved around the Y axis. To do this I am trying to integrate only one quadrant because the expansion from that is trivial. To do this I am using the following code:

$Assumptions=0<x<1 && \[Alpha]>1 && \[Beta] >1&&x<R<1;
dy=D[Sqrt[R^2-x^2],R]
CurveFunc=dy*R^(\[Alpha]-1)*(1-R)^(\[Beta]-1)/Beta[\[Alpha],\[Beta]];
CurvePDF=Expand[Integrate[CurveFunc,{R,x,1} ]]

This produces the following Integral with primarily made up with hypergeometric functions:

enter image description here

But a quick review of that function and you will see that it is undefined or singular at Integer values of [Alpha]. Which is incorrect because if you substitute in an integer value of [Alpha] before hand it will produce a valid return ie:

CurvePDF=Expand[Integrate[CurveFunc/.\[Alpha]->6,{R,x,1} ]]

Will return a function that is continuous. I would like to find the generic integral that works for all real Alpha > 1. Does anyone have an idea how to get Mathematica to produce an integral without those singularities?

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  • $\begingroup$ Typically, solutions of this sort are "generically correct"; that is, they are valid except on a countable subset of parameter values. Unfortunately, those values (e.g. the integers) are often the values of interest in applications. $\endgroup$ – J. M.'s ennui♦ Apr 11 '17 at 13:44
  • $\begingroup$ I have tried including the assumption that Alpha is an integer but it still produces the same always invalid result for that version. $\endgroup$ – N D Apr 11 '17 at 13:49
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When you calculate the integral for certain values of alpha and have a close look at the results, you can extract a general rule for Integer-alphas:

 int11[r_ /; r > 1 && EvenQ[r]] = (-1)^(r/2) 1/
      Beta[r, \[Beta]] 2^-\[Beta] Sqrt[\[Pi]] Gamma[\[Beta]] MeijerG[{{1/
      2}, {(r + \[Beta])/2, (r + 1 + \[Beta])/2}}, {{0, 
      r/2}, {(r + 1)/2}}, x^2]

 int11[r_ /; r > 1 && OddQ[r]] = (-1)^((r - 1)/2) 1/
      Beta[r, \[Beta]] 2^-\[Beta] Sqrt[\[Pi]] Gamma[\[Beta]] MeijerG[{{1/
      2}, {(r + 1)/2 + \[Beta]/2, (r + \[Beta])/2}}, {{0, (r + 1)/
      2}, {r/2}}, x^2]

Here the test

    Table[
          int11[j] == 
          Expand[Integrate[CurveFunc /. \[Alpha] -> j, {R, x, 1}]] // 
          Simplify, {j, 2, 15}]

    (*   {True, True, True, True, True, True, True, True, True, \
          True, True, True, True, True}    *)

    Plot[CurvePDF /. {\[Beta] -> 2.2, x -> .1}, {\[Alpha], 1, 15}, 
       PlotRange -> All, 
       Epilog -> 
       Point[Table[{\[Alpha], 
       int11[\[Alpha]] /. {\[Beta] -> 2.2, x -> .1}}, {\[Alpha], 2, 
       15}]], AxesLabel -> {\[Alpha], 
       "CurvePDF/.{\[Beta]\[Rule]2.2,x\[Rule].1}"}]

enter image description here

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