Integral could lead to Hypergeometric function


by DMESONS
Tags: function, hypergeometric, hypergeomtric, integral, lead
DMESONS
DMESONS is offline
#1
Nov8-13, 11:52 AM
P: 27
How can I perform this integral



\begin{equation}
\int^∞_a dq \frac{1}{(q+b)} (q^2-a^2)^n (q-c)^n ?
\end{equation}

all parameters are positive (a, b, and c) and n>0.

I tried using Mathemtica..but it doesn't work!

if we set b to zero, above integral leads to the hypergeometric function!
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JJacquelin
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#2
Nov8-13, 12:16 PM
P: 744
Hi !

What do you think about the convergence, or not ?
DMESONS
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#3
Nov8-13, 12:41 PM
P: 27
Quote Quote by JJacquelin View Post
Hi !

What do you think about the convergence, or not ?
Thanks for your comment.

The integral is convergent for some values of (n) which can be either positive or negative.

As I mentioned, if I set b =0, the result have the form of hypergeometric function!

But if b is not zero, Mathematica can't solve it.

Mandelbroth
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#4
Nov8-13, 02:09 PM
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P: 597

Integral could lead to Hypergeometric function


Quote Quote by DMESONS View Post
How can I perform this integral



\begin{equation}
\int^∞_a dq \frac{1}{(q+b)} (q^2-a^2)^n (q-c)^n ?
\end{equation}

all parameters are positive (a, b, and c) and n>0.

I tried using Mathemtica..but it doesn't work!

if we set b to zero, above integral leads to the hypergeometric function!
I'll preface this by asking what it is for, but I'll try to provide a partial solution, too.

To start, what on Earth is this for? We do we come up with such silly things to integrate?

Second, let's see if we can simplify things considerably:

$$\int\limits_{[a,+\infty)}\left(\frac{1}{q+b}(q^2-a^2)^n(q-c)^n\right)\mathrm{d}q=\int\limits_{[a,+\infty)}\left(\frac{1}{q+b}(q-a)^n(q+a)^n(q-c)^n\right)\mathrm{d}q.$$

I'm thinking we might just approach this by means of partial fractions. On cursory examination, I don't see a contour that would simplify things, so brute force might be necessary.
JJacquelin
JJacquelin is offline
#5
Nov9-13, 02:10 AM
P: 744
Quote Quote by DMESONS View Post
all parameters are positive (a, b, and c) and n>0.
Quote Quote by DMESONS View Post
The integral is convergent for some values of (n) which can be either positive or negative.
Hi !
Would you mind give a non contradictory wording of the question about the sign of n.
JJacquelin
JJacquelin is offline
#6
Nov10-13, 02:17 AM
P: 744
Quote Quote by DMESONS View Post
The integral is convergent for some values of (n) which can be either positive or negative.
Hi !
For which value of (n) the integral is convergent ?
Clue : The integral is NOT convergent for any integer (n), either positive or negative or n=0.


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