Please help me to solve the integral:
$$\int_0^{-2w}\phi^{-k-\frac{1}{2}}\frac{{_2F_1(k-\frac{1}{2},k,k+\frac{1}{2};1-\frac{k-\frac{3}{2}}{\phi})}}{\sqrt{-\phi-2\omega}}$$ (see the attachment)
I have solved this in Mathematica.but I am not able to way a
general result even in the table of integrals. Please help me out. Thanks in advance
k, w, are constants, 2F1 is the hypergeometric function...
It does not in converge in (-2*w), Mathematica said.
I have tried simple integer values for w and k, 1 and 2, and the integrand reduces to elementary funcions an it is clearly seen that the definite integra is divergent.
Here is an image attached and the latex code
In[5]:= \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\(-2\)*2\)]\(
\*SuperscriptBox[\(\[Phi]\), \(\(-1\) -
\*FractionBox[\(1\), \(2\)]\)]*
\*FractionBox[\(Hypergeometric2F1[1 -
\*FractionBox[\(1\), \(2\)], 1, 1 +
\*FractionBox[\(1\), \(2\)], 1 -
\*FractionBox[\(2 -
\*FractionBox[\(2\), \(3\)]\), \(\[Phi]\)]]\),
SqrtBox[\(\(-\[Phi]\) - 2*2\)]] \[DifferentialD]\[Phi]\)\)
Out[5]= \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(-4\)]\(
\*FractionBox[\(ArcTanh[
\*SqrtBox[\(1 -
\*FractionBox[\(4\), \(3\ \[Phi]\)]\)]]\), \(
\*SqrtBox[\(1 -
\*FractionBox[\(4\), \(3\ \[Phi]\)]\)]\
\*SqrtBox[\(\(-4\) - \[Phi]\)]\
\*SuperscriptBox[\(\[Phi]\), \(3/2\)]\)] \[DifferentialD]\[Phi]\)\)
Prudnikov, Brychkov, Marichev, Integrals and Series, vol.3, 2.21.1.14 after change of variable fi->1/fi
Hi Harikrishnan, Thank you for your post.
if you have solved your problem using mathematica, please write the answer.
Please make my understanding clear about the function 2F1.
Please discuss about the singularities in the integrand if any? or how many? and what type?
I have no idea, I like to learn from you.
I understand from your replies that Fye is the integration variable on which you are integrating although you forget to write seeing the expression 2F1.
Harikrishnan,
please write you message here rather than sending a private message. I am happy to get your reply that you could not solve it using Mathematica.
In reply to my question, you have sent a link of a pdf file about hypergeometric function. Thank you for it. It is ok for them who are interested only in hypergeometric function.
Please note that here our interest is to perform the integration. So please take a handbook and find the series of 2F1. Please find the crietria of convergence of 2F1 function.
Next look at other terms in the integral and singularities.
I think it will help you.
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