Difficult integral involving hypergeometric function

[bruno67]

I am trying to calculate the following integral

I=\int_0^\infty\frac{x}{(x+ia)^2} {\mbox{$_2$F$_1\!$}}\left(\frac{1}{2},b,\frac{3}{2},-\frac{x^2}{c^2}\right) dx.<br />

I tried several different ways but drew a complete blank. Is there a way of converting this nasty hypergeometric function into something which is more tractable? Unfortunately the definition of the hypergeometric function as a series cannot be used, since it is only valid when the fourth argument is less than 1 in modulus.

Thanks.

 

[3.1415926535]

Question: Why does x need to be less than 1? If it is more than 1, then the series diverges?

 

[bruno67]

Question: Why does x need to be less than 1? If it is more than 1, then the series diverges?

Yes, if the fourth argument is equal to 1 or larger, the series diverges. In that case, the hypergeometric function can be defined by analytic continuation.

 

[3.1415926535]

If there is no way to write the Hypergeometric function as a series(for any x), then I don't see how you can transform it something else to integrate it. Sorry

Here is everything about the Hypergeometric function of the 1st kind
http://mathworld.wolfram.com/HypergeometricFunction.html
I hope you find it usefull.

One last thing: The Hypergeometric functions are used to solve ODEs with variable coefficients. If you can construct the ODE and solve it using Frobenius series, then is there any possibility that the series you will find will converge for every x?

 

[JJacquelin]

Hi !
Sorry, I cannot read what you have written.
Is it possible to display your integral on another way ?
Look at what I see :

 

[3.1415926535]

Hi !
Sorry, I cannot read what you have written.
Is it possible to display your integral on another way ?
Look at what I see :

\begin{array}\\I=\int_0^\infty\frac{x}{(x+ia)^2} {\mbox{$_2$F$_1\!$}}\left(\frac{1}{2},b,\frac{3}{2 },-\frac{x^2}{c^2}\right) \end{array}

Better now?
If it is the same, copy the LaTex code and paste it here
http://www.codecogs.com/latex/eqneditor.php

 

[fluidistic]

Hi !
Sorry, I cannot read what you have written.
Is it possible to display your integral on another way ?
Look at what I see :

Right click on what you see and select "reload" or something like that, depending upon your internet browser. It's supposed to be an image and it hasn't loaded in your case.

 

[bruno67]

Hi !
Sorry, I cannot read what you have written.
Is it possible to display your integral on another way ?
Look at what I see :

Here is the TeX source:

$I=\int_0^\infty\frac{x}{(x+ia)^2} {\mbox{$_2$F$_1\!$}}\left(\frac{1}{2},b,\frac{3}{2},-\frac{x^2}{c^2}\right) dx.$

You can also visualize it by clicking "Quote" at the end of a message.

 

[JJacquelin]

I realize that LATEX no longer works for all the Physics Forums.
It is impossible for me to read the questions which include formulas.
Sorry, until the problem be solved, this draw me to quit the Physics Forums.
Bye-bye !

 

[fluidistic]

I realize that LATEX no longer works for all the Physics Forums.
It is impossible for me to read the questions which include formulas.
Sorry, until the problem be solved, this draw me to quit the Physics Forums.
Bye-bye !

Latex works for me. :/

 

[JJacquelin]

First, I would like thank 3.1415926535 , fluidistic and bruno67 for their advices. But I didn't succeed to make LATEX work again.
What is strange is that old topics, where LATEX was correctly working, are now illegible.
The trouble is with Physics Forums only. Other sites and forums with LATEX are properly working.
Well, fluidistic gave the image .jpg of the integral. This enable me to answer :
In the particular case, the hypergeometric function reduces to Incomplete Beta Function, which is a function of lower level (in attachment)
But, even on the form of Incomplete Beta, the whole integal seems very difficult to be analytically solved.

 

[Fredrik]

JJacquelin, check out this thread, in particular post #165.

 

[JJacquelin]

JJacquelin, check out this thread, in particular post #165.

Thank you for your help. I made the test. The result is that Javasript is correct. That is not surprising anyways since their is no problem with other forums than Physics Forums.

 

[Fredrik]

OK, then I don't know what to do. Perhaps try another browser? You might want to keep an eye on that thread. Someone might post a solution there. You could also try posting more details about your problem there.

 

[JJacquelin]

OK, then I don't know what to do. Perhaps try another browser? You might want to keep an eye on that thread. Someone might post a solution there. You could also try posting more details about your problem there.

I kept an eye on that thread and on some other threads. To my surprise I just see that LATEX is working again ! and yet I did nothing...