Double integral with logarithms [on hold]












3












$begingroup$


How to solve this integral:



$$Jequiv int_{0}^{1}{int_{0}^{1}{frac{ln x-ln y}{x-y}}}dxdy$$



I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$frac{{{pi }^{2}}}{3}.$$



Thanks!.






integration







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Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$



put on hold as off-topic by YCor, user44191, Pace Nielsen, Gerald Edgar, Piotr Hajlasz 5 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Pace Nielsen, Gerald Edgar, Piotr Hajlasz

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 4




    $begingroup$
    MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
    $endgroup$
    – user64494
    11 hours ago






  • 7




    $begingroup$
    I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
    $endgroup$
    – RP_
    10 hours ago






  • 3




    $begingroup$
    @user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
    $endgroup$
    – Fedor Petrov
    8 hours ago






  • 2




    $begingroup$
    @user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
    $endgroup$
    – Fedor Petrov
    5 hours ago






  • 1




    $begingroup$
    Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
    $endgroup$
    – YCor
    3 hours ago
















3












$begingroup$


How to solve this integral:



$$Jequiv int_{0}^{1}{int_{0}^{1}{frac{ln x-ln y}{x-y}}}dxdy$$



I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$frac{{{pi }^{2}}}{3}.$$



Thanks!.






integration







share|cite|improve this question









New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$



put on hold as off-topic by YCor, user44191, Pace Nielsen, Gerald Edgar, Piotr Hajlasz 5 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Pace Nielsen, Gerald Edgar, Piotr Hajlasz

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 4




    $begingroup$
    MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
    $endgroup$
    – user64494
    11 hours ago






  • 7




    $begingroup$
    I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
    $endgroup$
    – RP_
    10 hours ago






  • 3




    $begingroup$
    @user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
    $endgroup$
    – Fedor Petrov
    8 hours ago






  • 2




    $begingroup$
    @user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
    $endgroup$
    – Fedor Petrov
    5 hours ago






  • 1




    $begingroup$
    Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
    $endgroup$
    – YCor
    3 hours ago














3












3








3


1



$begingroup$


How to solve this integral:



$$Jequiv int_{0}^{1}{int_{0}^{1}{frac{ln x-ln y}{x-y}}}dxdy$$



I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$frac{{{pi }^{2}}}{3}.$$



Thanks!.






integration







share|cite|improve this question









New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




How to solve this integral:



$$Jequiv int_{0}^{1}{int_{0}^{1}{frac{ln x-ln y}{x-y}}}dxdy$$



I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$frac{{{pi }^{2}}}{3}.$$



Thanks!.






integration




integration






share|cite|improve this question









New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









YCor

29.1k486140




29.1k486140






New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 11 hours ago









Jesús Álvarez LoboJesús Álvarez Lobo

251




251




New contributor




Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Jesús Álvarez Lobo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




put on hold as off-topic by YCor, user44191, Pace Nielsen, Gerald Edgar, Piotr Hajlasz 5 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Pace Nielsen, Gerald Edgar, Piotr Hajlasz

If this question can be reworded to fit the rules in the help center, please edit the question.







put on hold as off-topic by YCor, user44191, Pace Nielsen, Gerald Edgar, Piotr Hajlasz 5 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Pace Nielsen, Gerald Edgar, Piotr Hajlasz

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 4




    $begingroup$
    MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
    $endgroup$
    – user64494
    11 hours ago






  • 7




    $begingroup$
    I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
    $endgroup$
    – RP_
    10 hours ago






  • 3




    $begingroup$
    @user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
    $endgroup$
    – Fedor Petrov
    8 hours ago






  • 2




    $begingroup$
    @user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
    $endgroup$
    – Fedor Petrov
    5 hours ago






  • 1




    $begingroup$
    Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
    $endgroup$
    – YCor
    3 hours ago














  • 4




    $begingroup$
    MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
    $endgroup$
    – user64494
    11 hours ago






  • 7




    $begingroup$
    I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
    $endgroup$
    – RP_
    10 hours ago






  • 3




    $begingroup$
    @user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
    $endgroup$
    – Fedor Petrov
    8 hours ago






  • 2




    $begingroup$
    @user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
    $endgroup$
    – Fedor Petrov
    5 hours ago






  • 1




    $begingroup$
    Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
    $endgroup$
    – YCor
    3 hours ago








4




4




$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
$endgroup$
– user64494
11 hours ago




$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ frac{pi ^2}{3}$.
$endgroup$
– user64494
11 hours ago




7




7




$begingroup$
I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
$endgroup$
– RP_
10 hours ago




$begingroup$
I do not think this is such a bad question. I would not expect even every research mathematician to come up with the answer right away. So I vote not to close.
$endgroup$
– RP_
10 hours ago




3




3




$begingroup$
@user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
$endgroup$
– Fedor Petrov
8 hours ago




$begingroup$
@user64494 what is a general strategy: when you have an integral which you do not know how to calculate, how should you decide between MSE and MO?
$endgroup$
– Fedor Petrov
8 hours ago




2




2




$begingroup$
@user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
$endgroup$
– Fedor Petrov
5 hours ago




$begingroup$
@user64494 being the author of several papers devoted to exact closed-form integration, I feel so old now.
$endgroup$
– Fedor Petrov
5 hours ago




1




1




$begingroup$
Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
$endgroup$
– YCor
3 hours ago




$begingroup$
Closing a question doesn't mean it's bad. Also doesn't mean it would be solved right away by any research mathematician.
$endgroup$
– YCor
3 hours ago










1 Answer
1






active

oldest

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9












$begingroup$

By symmetry we have $J/2=int_0^1 dx int_0^x f(x, y) dy$ where $f(x, y) $ is your integrand. Integrating against $y$ for fixed $x$ we denote $y=tx$, $t$ varies from 0 to 1 and the integral against $y$ reads as $-int_0^1 frac{log t} {1-t}dt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac{1}{1 - t} =sum_{n>0 } t^{n-1}$ and integrate term-wise to get $sum 1/n^2$).






share|cite|improve this answer











$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    9












    $begingroup$

    By symmetry we have $J/2=int_0^1 dx int_0^x f(x, y) dy$ where $f(x, y) $ is your integrand. Integrating against $y$ for fixed $x$ we denote $y=tx$, $t$ varies from 0 to 1 and the integral against $y$ reads as $-int_0^1 frac{log t} {1-t}dt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac{1}{1 - t} =sum_{n>0 } t^{n-1}$ and integrate term-wise to get $sum 1/n^2$).






    share|cite|improve this answer











    $endgroup$


















      9












      $begingroup$

      By symmetry we have $J/2=int_0^1 dx int_0^x f(x, y) dy$ where $f(x, y) $ is your integrand. Integrating against $y$ for fixed $x$ we denote $y=tx$, $t$ varies from 0 to 1 and the integral against $y$ reads as $-int_0^1 frac{log t} {1-t}dt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac{1}{1 - t} =sum_{n>0 } t^{n-1}$ and integrate term-wise to get $sum 1/n^2$).






      share|cite|improve this answer











      $endgroup$
















        9












        9








        9





        $begingroup$

        By symmetry we have $J/2=int_0^1 dx int_0^x f(x, y) dy$ where $f(x, y) $ is your integrand. Integrating against $y$ for fixed $x$ we denote $y=tx$, $t$ varies from 0 to 1 and the integral against $y$ reads as $-int_0^1 frac{log t} {1-t}dt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac{1}{1 - t} =sum_{n>0 } t^{n-1}$ and integrate term-wise to get $sum 1/n^2$).






        share|cite|improve this answer











        $endgroup$



        By symmetry we have $J/2=int_0^1 dx int_0^x f(x, y) dy$ where $f(x, y) $ is your integrand. Integrating against $y$ for fixed $x$ we denote $y=tx$, $t$ varies from 0 to 1 and the integral against $y$ reads as $-int_0^1 frac{log t} {1-t}dt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac{1}{1 - t} =sum_{n>0 } t^{n-1}$ and integrate term-wise to get $sum 1/n^2$).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 10 hours ago









        T. Amdeberhan

        18.4k230132




        18.4k230132










        answered 11 hours ago









        Fedor PetrovFedor Petrov

        52.3k6122239




        52.3k6122239















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