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I want to solve the following integration $$\int_0^{\infty}[1-(\frac{1}{1+x})^M]x^{-\frac{2}{\alpha}-1}dx$$ where $M$ is a positive integer and $\alpha \geq 2$

My attempt:

In my attempt I use the Binomial expansion for $(1+x)^{-M}$ and the result is as follows $$\int_0^{\infty}[Mx-\frac{1}{2}M(M+1)x^2-\frac{1}{6}M(M+1)(M+2)x^3+\cdots]x^{-\frac{2}{\alpha}-1}dx$$ As can be seen that the result will come out to be zero because $\infty^{-j}=0$ and similarly $0^i=0$. Your help in solving this integration will be much appreciated. Thanks in advance.

Harry Peter
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Frank Moses
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1 Answers1

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You correctly pointed out the problem if the binomial expansion is used.

If fact, you need to use hypergeometric functions since $$\int\left(1-\frac{1}{(1+x)^M}\right)x^{-\frac{2}{\alpha}-1}\,dx=\frac{1}{2} \alpha x^{-2/\alpha} \left(\, _2F_1\left(-\frac{2}{\alpha},M;1-\frac{2}{\alpha};-x\right)-1\right)$$ Using a CAS, what was obtained is $$\int_0^\infty\left(1-\frac{1}{(1+x)^M}\right)x^{-\frac{2}{\alpha}-1}\,dx=-\frac{\Gamma \left(-\frac{2}{\alpha}\right) \Gamma \left(M+\frac{2}{\alpha}\right)}{\Gamma (M)}$$

Claude Leibovici
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    The result I gave is positive. Plot function $\Gamma \left(-\frac{2}{z}\right)$ for $2 \leq z \leq 10$; it is always negative (even going through a maximum). Try with $\alpha=4$ and $M=2$; you should get $\frac{3 \pi }{2}$. – Claude Leibovici Jul 28 '16 at 05:28
  • I think the result provided in eq(43) of http://arxiv.org/abs/1405.2013 (with the value of $k_1$ provided in eq(18)) is wrong. because based on your answer the value of $k_1$ should be $\pi q_c^D \lambda_B(aP_B)^{\frac{2}{\alpha}}\Gamma(1-\frac{2}{\alpha})\frac{\Gamma(|C|+\frac{2}{\alpha})}{\Gamma(|C|)}$ – Frank Moses Jul 28 '16 at 05:42
  • its $\frac{2}{\alpha}$ in my above comment where it is written in red. – Frank Moses Jul 28 '16 at 05:45
  • yes I agree that its positive actually if we use the property $\Gamma(1-x)=-x\Gamma(-x)$ then we could see that both (yours and that I wrote in my deleted comment )result were exactly same. But I am not sure if the result provided in the above paper is right until unless there is some property like this http://math.stackexchange.com/questions/1873583/is-it-true-to-write. Thank you for your help. – Frank Moses Jul 28 '16 at 05:54
  • Good to see that we agree ! You are very welcome. – Claude Leibovici Jul 28 '16 at 06:10