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Question Number 159447 by mnjuly1970 last updated on 17/Nov/21

$$$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{\Omega }:={\int }_0^1{\int }_0^1\frac{x^{\frac{t}{2}}-x^t}{1-x}dxdt=?$$$$---m.n---$$

Commented by Ar Brandon last updated on 17/Nov/21

Answered by Ar Brandon last updated on 17/Nov/21

$$\mathrm{\Omega }={\int }_0^1{\int }_0^1\frac{x^{\frac{t}{2}}-x^t}{1-x}dxdt$$$$={\int }_0^1(\psi (t+1)-\psi (\frac{t}{2}+1))dt$$$$={[\ln \left(\mathrm{\Gamma }(t+1)\right)-2\ln \left(\mathrm{\Gamma }(\frac{t}{2}+1)\right)]}_0^1$$$$=\ln \left(\frac{\mathrm{\Gamma }\left(2\right)}{{\mathrm{\Gamma }}^2\left(\frac{3}{2}\right)}\right)-\ln \left(\frac{\mathrm{\Gamma }\left(1\right)}{{\mathrm{\Gamma }}^2\left(1\right)}\right)=\ln \left(\frac{4}{\pi }\right)$$

Commented by mnjuly1970 last updated on 17/Nov/21

$$peacebeuponyousirbrandon..$$

Commented by Ar Brandon last updated on 17/Nov/21

With you too, Sir.

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