Difficult integral involving hypergeometric function

In summary, this conversation discusses the difficulty of integrating a hypergeometric function with a fourth argument greater than or equal to 1, the possibility of defining it through analytic continuation, and the use of hypergeometric functions in solving ODEs. There is also mention of the issue with using LaTeX on Physics Forums, which has since been resolved.
  • #1
bruno67
32
0
I am trying to calculate the following integral

[tex]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.
[/tex]

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.
 
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  • #2
Question: Why does x need to be less than 1? If it is more than 1, then the series diverges?
 
  • #3
3.1415926535 said:
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.
 
  • #4
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?
 
  • #5
Hi !
Sorry, I cannot read what you have written.
Is it possible to display your integral on another way ?
Look at what I see :
 

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  • #6
JJacquelin said:
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
 
  • #7
JJacquelin said:
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.
 
  • #8
JJacquelin said:
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.
 
  • #9
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 !
 
  • #10
JJacquelin said:
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. :/
 

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  • #11
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.
 

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Last edited:
  • #12
JJacquelin, check out this thread, in particular post #165.
 
  • #13
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.
 

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  • #14
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.
 
  • #15
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...
 

1. What is a difficult integral involving hypergeometric function?

A difficult integral involving hypergeometric function is an integration problem that involves the use of the hypergeometric function, which is a special mathematical function that appears in many areas of mathematics, physics, and engineering. It is known for its complicated and challenging nature, making it a common topic of interest for scientists and mathematicians.

2. How is the hypergeometric function used in integrals?

The hypergeometric function, denoted as $_2F_1$, is used in integrals as a solution to many types of integrals involving polynomials, logarithms, and other functions. It is also used as a tool for evaluating integrals with complex or difficult functions that cannot be solved using traditional methods.

3. What makes these integrals difficult?

The complexity of these integrals comes from the properties and behavior of the hypergeometric function itself. It has a wide range of parameters, making it challenging to find a specific solution for each case. Additionally, the integrals may involve multiple terms and require advanced mathematical techniques to solve.

4. What are some applications of difficult integrals involving hypergeometric function?

Difficult integrals involving hypergeometric function have various applications in physics, engineering, and statistics. In physics, it is used to solve problems related to quantum mechanics, statistical mechanics, and general relativity. In engineering, it is used to model and analyze complex systems. In statistics, it is used in probability distributions and statistical analysis.

5. How can I approach solving a difficult integral involving hypergeometric function?

Solving difficult integrals involving hypergeometric function requires a good understanding of the properties and behavior of the hypergeometric function. It also requires knowledge of advanced mathematical techniques, such as contour integration and series expansions. It is recommended to consult with a mathematician or use specialized software to solve these types of integrals.

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