Mixing
Need help about references for 2D delta “function”
$begingroup$
I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true:
$$left(
frac{partial^2}{partial x^2}
+
frac{partial^2}{partial y^2}
right)
log{left(x^2+y^2right)^{frac{1}{2}}}
=
2pidelta(x)delta(y).$$
Function $log$ is natural logarithm, i.e. $log e=1$, and $delta(x)$ and $delta(y)$ are Dirac's delta "functions". Unfortunately, I cant remember where in the literature I have seen this equation since the last time I was using it was in 2005. For that reason, I kindly ask if someone can point me to the papers or books which I can use for reference.
multivariable-calculus partial-derivative dirac-delta laplacian
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
$endgroup$
migrated from physics.stackexchange.com Jan 22 at 22:50
This question came from our site for active researchers, academics and students of physics.
add a comment |
$begingroup$
I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true:
$$left(
frac{partial^2}{partial x^2}
+
frac{partial^2}{partial y^2}
right)
log{left(x^2+y^2right)^{frac{1}{2}}}
=
2pidelta(x)delta(y).$$
Function $log$ is natural logarithm, i.e. $log e=1$, and $delta(x)$ and $delta(y)$ are Dirac's delta "functions". Unfortunately, I cant remember where in the literature I have seen this equation since the last time I was using it was in 2005. For that reason, I kindly ask if someone can point me to the papers or books which I can use for reference.
multivariable-calculus partial-derivative dirac-delta laplacian
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
$endgroup$
migrated from physics.stackexchange.com Jan 22 at 22:50
This question came from our site for active researchers, academics and students of physics.
1
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50
add a comment |
$begingroup$
I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true:
$$left(
frac{partial^2}{partial x^2}
+
frac{partial^2}{partial y^2}
right)
log{left(x^2+y^2right)^{frac{1}{2}}}
=
2pidelta(x)delta(y).$$
Function $log$ is natural logarithm, i.e. $log e=1$, and $delta(x)$ and $delta(y)$ are Dirac's delta "functions". Unfortunately, I cant remember where in the literature I have seen this equation since the last time I was using it was in 2005. For that reason, I kindly ask if someone can point me to the papers or books which I can use for reference.
multivariable-calculus partial-derivative dirac-delta laplacian
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
$endgroup$
I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true:
$$left(
frac{partial^2}{partial x^2}
+
frac{partial^2}{partial y^2}
right)
log{left(x^2+y^2right)^{frac{1}{2}}}
=
2pidelta(x)delta(y).$$
Function $log$ is natural logarithm, i.e. $log e=1$, and $delta(x)$ and $delta(y)$ are Dirac's delta "functions". Unfortunately, I cant remember where in the literature I have seen this equation since the last time I was using it was in 2005. For that reason, I kindly ask if someone can point me to the papers or books which I can use for reference.
multivariable-calculus partial-derivative dirac-delta laplacian
multivariable-calculus partial-derivative dirac-delta laplacian
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
edited Jan 23 at 8:49
Martin Sleziak
44.8k10119273
44.8k10119273
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
asked Jan 22 at 21:20
gallieo1985gallieo1985
64
64
migrated from physics.stackexchange.com Jan 22 at 22:50
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Jan 22 at 22:50
This question came from our site for active researchers, academics and students of physics.
1
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50
add a comment |
1
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50
1
1
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)
This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function
You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
$endgroup$
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)
This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function
You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
$endgroup$
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
add a comment |
$begingroup$
The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)
This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function
You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
$endgroup$
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
add a comment |
$begingroup$
The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)
This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function
You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
$endgroup$
The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)
This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function
You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
answered Jan 22 at 22:47
Zecheng GanZecheng Gan
11
11
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
add a comment |
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
Yes I know its standard, however I am having trouble finding citable reference in standard books
$endgroup$
– gallieo1985
Jan 23 at 0:33
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
$begingroup$
I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential.
$endgroup$
– Zecheng Gan
Jan 23 at 1:07
add a comment |
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1
$begingroup$
Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs
$endgroup$
– Aaron Stevens
Jan 22 at 21:23
$begingroup$
Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice
$endgroup$
– gallieo1985
Jan 22 at 21:29
$begingroup$
Math mods: Please merge.
$endgroup$
– Qmechanic
Jan 22 at 22:50