Double integral with logarithms [on hold]
$begingroup$
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
edited 3 hours ago
YCor
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
$endgroup$
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.
|
show 2 more comments
$begingroup$
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
edited 3 hours ago
YCor
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
$endgroup$
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
|
show 2 more comments
$begingroup$
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
edited 3 hours ago
YCor
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
$endgroup$
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
edited 3 hours ago
YCor
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
edited 3 hours ago
YCor
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
edited 3 hours ago
YCor
29.1k486140
edited 3 hours ago
YCor
29.1k486140
edited 3 hours ago
YCor
29.1k486140
29.1k486140
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
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.
asked 11 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
251
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
|
show 2 more comments
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
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
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 $sum 1/n^2$).
edited 10 hours ago
T. Amdeberhan
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $sum 1/n^2$).
edited 10 hours ago
T. Amdeberhan
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
$endgroup$
add a comment |
$begingroup$
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 $sum 1/n^2$).
edited 10 hours ago
T. Amdeberhan
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
$endgroup$
add a comment |
$begingroup$
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 $sum 1/n^2$).
edited 10 hours ago
T. Amdeberhan
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
$endgroup$
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 $sum 1/n^2$).
edited 10 hours ago
T. Amdeberhan
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
edited 10 hours ago
T. Amdeberhan
18.4k230132
edited 10 hours ago
T. Amdeberhan
18.4k230132
edited 10 hours ago
T. Amdeberhan
18.4k230132
18.4k230132
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
answered 11 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
52.3k6122239
add a comment |
add a comment |
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