Cannot do the integral of the Hyper-geometric function?

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SUMMARY

The discussion centers on the inability of Mathematica to compute the integral of the hypergeometric function ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##, while it successfully evaluates ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##. Participants conclude that the second integral likely cannot be solved analytically. They recommend consulting Rydzik and Gradshteyn for further insights on this topic.

PREREQUISITES
  • Understanding of hypergeometric functions, specifically ##_2F_1##.
  • Familiarity with Mathematica version 12.3 or later for computational capabilities.
  • Knowledge of integral calculus, particularly improper integrals.
  • Access to reference texts such as Rydzik and Gradshteyn for advanced mathematical functions.
NEXT STEPS
  • Research the properties and applications of hypergeometric functions.
  • Explore advanced integral techniques in Mathematica.
  • Study the reference material by Rydzik and Gradshteyn for deeper insights into complex integrals.
  • Investigate numerical methods for approximating integrals that cannot be solved analytically.
USEFUL FOR

Mathematicians, researchers in applied mathematics, and anyone interested in advanced integral calculus and hypergeometric functions.

Chenkb
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Dear friends:
It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##.
Are there any major differences between this two types? Is it possible to do the second kind of integral?
Best regards!

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I don't believe that that particular integral can be solved analytically. Best place to look is in Rydzik and Gradstien (sp)...
 
Dr Transport said:
I don't believe that that particular integral can be solved analytically. Best place to look is in Rydzik and Gradstien (sp)...
Many thanks! That reference is excellent!
 

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