$\int_{0}^{\infty} \log_{2}(1 + x \rho)\,x^{\left(\tfrac{\alpha \mu}{2} - 1\right)}\, \exp \left( -\mu \Phi x^{\alpha/2} \right) \, dx $

Asked Feb 08 '18 at 18:28

Active Feb 08 '18 at 20:48

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While solving one of the problems, I came across the following integral: $$I = \int_{0}^{\infty} \log_{2}(1 + x \rho)\,x^{\left(\tfrac{\alpha \mu}{2} - 1\right)}\, \exp \left( -\mu \Phi x^{\alpha/2} \right) \, dx $$

Here $\alpha > 0$, $\mu \in \mathbb{Z}^+$, $\Phi, x, \rho \in \mathbb{R}^+$. Can anyone please help me find an exact solution or a tight lower bound (when $\rho \gg 1$) for the above integration?

PS: Even if someone can tell the final answer with a proper reference, that would be very helpful.

edited Feb 08 '18 at 20:48

asked Feb 08 '18 at 18:28

Nash J.

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Nash, I did calculate similar integrals in the past, mostly powers of the log-function. The integral can be solved for some constant $\alpha$ immediately, the general case is more tricky. The best approach is to use the Mellin transformation and the Plancherel theorem [https://math.stackexchange.com/questions/334775/strategy-for-improper-integrals-related-to-the-beta-function-2/334936#334936]. This should work and the only challenge is to find the inverse Transform. – stocha Feb 08 '18 at 19:46
@stocha: Thanks a million for your comment. I am looking into this! – Nash J. Feb 08 '18 at 20:49

$\int_{0}^{\infty} \log_{2}(1 + x \rho)\,x^{\left(\tfrac{\alpha \mu}{2} - 1\right)}\, \exp \left( -\mu \Phi x^{\alpha/2} \right) \, dx $

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