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Two special logarithmic integral

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Two special logarithmic integral

Sun Oct 12, 2014 1:18 pm

Posts: 105
$$\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)L{i_3}\left( {\frac{{1 + x}}{2}} \right)}}{x}dx} = ?,\int\limits_0^1 {\frac{{{{\ln }^2}\left( {1 - x} \right)L{i_2}\left( {\frac{{1 + x}}{2}} \right)}}{x}dx} = ?.$$

Re: Two special logarithmic integral

Sun Oct 12, 2014 6:23 pm

Posts: 4

$\displaystyle{\large\int}_0^1\frac{\ln(1-x)\,\operatorname{Li}_3\left(\frac{1+x}2\right)}xdx=\frac{29\,\zeta(5)}{16}-\frac{19\pi^2}{96}\zeta(3)+\frac{5\,\zeta(3)}{16}\ln^22+\frac{\ln^52}{40}-\frac{5\pi^2}{72}\ln^32+\frac{11\pi^4}{1440}\ln2-3\operatorname{Li}_5\left(\tfrac12\right).$

$\displaystyle{\large\int}_0^1\frac{\ln^2(1-x)\,\operatorname{Li}_2\left(\frac{1+x}2\right)}xdx=\frac{81\,\zeta(5)}{32}+\frac{5\pi^2}{16}\zeta(3)-\frac{\zeta(3)}8\ln^22+\frac1{15}\ln^52-\frac{\pi^2}{18}\ln^32-\frac{\pi^4}{15}\ln2+2\operatorname{Li}_5\left(\tfrac12\right)+2\operatorname{Li}_4\left(\tfrac12\right)\ln2.$

Re: Two special logarithmic integral

Sun Oct 12, 2014 7:04 pm

Posts: 85
Incredible!

Re: Two special logarithmic integral

Sun Oct 12, 2014 11:20 pm

Posts: 105
CleoMSE wrote:
$\displaystyle{\large\int}_0^1\frac{\ln(1-x)\,\operatorname{Li}_3\left(\frac{1+x}2\right)}xdx=\frac{29\,\zeta(5)}{16}-\frac{19\pi^2}{96}\zeta(3)+\frac{5\,\zeta(3)}{16}\ln^22+\frac{\ln^52}{40}-\frac{5\pi^2}{72}\ln^32+\frac{11\pi^4}{1440}\ln2-3\operatorname{Li}_5\left(\tfrac12\right).$

$\displaystyle{\large\int}_0^1\frac{\ln^2(1-x)\,\operatorname{Li}_2\left(\frac{1+x}2\right)}xdx=\frac{81\,\zeta(5)}{32}+\frac{5\pi^2}{16}\zeta(3)-\frac{\zeta(3)}8\ln^22+\frac1{15}\ln^52-\frac{\pi^2}{18}\ln^32-\frac{\pi^4}{15}\ln2+2\operatorname{Li}_5\left(\tfrac12\right)+2\operatorname{Li}_4\left(\tfrac12\right)\ln2.$

@CleoMSE.Do you have specific process？

Re: Two special logarithmic integral

Mon Oct 13, 2014 12:55 am

Posts: 24
If she were to tell you how she got that, she wouldn't be Cleo...

(She frequents Math Stack Exchange, and always gives her answers without telling how she got them.)

Re: Two special logarithmic integral

Sun Sep 11, 2016 4:41 am

Posts: 1
This is strange for anyone how the process of thinking is in her (cleo)mind