Double integral with logarithms Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)An interesting integral expression for $pi^n$?Integral involving Laguerre, Gaussian and modified Bessel functionhow to solve the rational exp integral?Nested trigonometric integralI want to disprove an equality involving a double integralIntegral of product of Gaussian pdf and cdfRewriting an elliptic integral in terms of theta functionsHow to solve this integral equation?Leibniz rule for Hadamard derivative?Integral of Classical Entropy
Double integral with logarithms
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)An interesting integral expression for $pi^n$?Integral involving Laguerre, Gaussian and modified Bessel functionhow to solve the rational exp integral?Nested trigonometric integralI want to disprove an equality involving a double integralIntegral of product of Gaussian pdf and cdfRewriting an elliptic integral in terms of theta functionsHow to solve this integral equation?Leibniz rule for Hadamard derivative?Integral of Classical Entropy
$begingroup$
How to solve this integral:
$$Jequiv int_0^1int_0^1fracln x-ln yx-ydxdy$$
I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$fracpi ^23.$$
Thanks!.
integral
edited 2 hours ago
user64494
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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.
$endgroup$
add a comment |
$begingroup$
How to solve this integral:
$$Jequiv int_0^1int_0^1fracln x-ln yx-ydxdy$$
I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$fracpi ^23.$$
Thanks!.
integral
edited 2 hours ago
user64494
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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.
$endgroup$
4
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
4
$begingroup$
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_
2 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
4 mins ago
add a comment |
$begingroup$
How to solve this integral:
$$Jequiv int_0^1int_0^1fracln x-ln yx-ydxdy$$
I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$fracpi ^23.$$
Thanks!.
integral
edited 2 hours ago
user64494
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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:
$$Jequiv int_0^1int_0^1fracln x-ln yx-ydxdy$$
I've tried it for polylogarithmic transformations, but I can not get the result, which I think should be $$fracpi ^23.$$
Thanks!.
integral
integral
edited 2 hours ago
user64494
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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 2 hours ago
user64494
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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 2 hours ago
user64494
1,818617
edited 2 hours ago
user64494
1,818617
edited 2 hours ago
user64494
1,818617
1,818617
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
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 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
asked 3 hours ago
Jesús Álvarez LoboJesús Álvarez Lobo
221
221
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.
4
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
4
$begingroup$
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_
2 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
4 mins ago
add a comment |
4
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
4
$begingroup$
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_
2 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
4 mins ago
4
4
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
4
4
$begingroup$
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_
2 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_
2 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
4 mins 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
4 mins ago
add a comment |
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 $-int_0^1 fraclog t 1-tdt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac11 - t =sum_n>0 t^n-1$ and integrate term-wise to get $sum 1/n^2$).
edited 1 hour ago
T. Amdeberhan
18.4k230132
answered 3 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
$endgroup$
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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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 $-int_0^1 fraclog t 1-tdt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac11 - t =sum_n>0 t^n-1$ and integrate term-wise to get $sum 1/n^2$).
edited 1 hour ago
T. Amdeberhan
18.4k230132
answered 3 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 $-int_0^1 fraclog t 1-tdt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac11 - t =sum_n>0 t^n-1$ and integrate term-wise to get $sum 1/n^2$).
edited 1 hour ago
T. Amdeberhan
18.4k230132
answered 3 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 $-int_0^1 fraclog t 1-tdt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac11 - t =sum_n>0 t^n-1$ and integrate term-wise to get $sum 1/n^2$).
edited 1 hour ago
T. Amdeberhan
18.4k230132
answered 3 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 $-int_0^1 fraclog t 1-tdt$. It does not depend on $x$ and is well known to be equal to $pi^2/6$ (you may use the geometric progression expansion $frac11 - t =sum_n>0 t^n-1$ and integrate term-wise to get $sum 1/n^2$).
edited 1 hour ago
T. Amdeberhan
18.4k230132
answered 3 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
edited 1 hour ago
T. Amdeberhan
18.4k230132
edited 1 hour ago
T. Amdeberhan
18.4k230132
edited 1 hour ago
T. Amdeberhan
18.4k230132
18.4k230132
answered 3 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
answered 3 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
answered 3 hours ago
Fedor PetrovFedor Petrov
52.3k6122239
52.3k6122239
add a comment |
add a comment |
Jesús Álvarez Lobo is a new contributor. Be nice, and check out our Code of Conduct.
Jesús Álvarez Lobo is a new contributor. Be nice, and check out our Code of Conduct.
Jesús Álvarez Lobo is a new contributor. Be nice, and check out our Code of Conduct.
Jesús Álvarez Lobo is a new contributor. Be nice, and check out our Code of Conduct.
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4
$begingroup$
MSE is a right place for such type questions. Both Maple and Mathematica confirm $ fracpi ^23$.
$endgroup$
– user64494
2 hours ago
4
$begingroup$
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_
2 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
4 mins ago