Neumann theorem for Bessel function of the first kind (arbitrary order)
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There is a Neumann theorem, proving that:
$J_0 big(sqrt{Z^2+z^2 - 2Zz cos phi} big)=sum_{k=0}^{infty} epsilon_k J_k(Z) J_k (z) cos k phi$
where $epsilon_0=1$, and $epsilon_k = 2$: $kgeq 1$. It is assumed that $sqrt{Z^2+z^2 - 2Zz cos phi} = |vec{Z} - vec{z}|$.
Does anyone know of a generalization for Bessel functions of arbitrary order $J_m big(sqrt{Z^2+z^2 - 2Zz cos phi} big)$: $m$ is any integer?
summation bessel-functions
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
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add a comment |
$begingroup$
There is a Neumann theorem, proving that:
$J_0 big(sqrt{Z^2+z^2 - 2Zz cos phi} big)=sum_{k=0}^{infty} epsilon_k J_k(Z) J_k (z) cos k phi$
where $epsilon_0=1$, and $epsilon_k = 2$: $kgeq 1$. It is assumed that $sqrt{Z^2+z^2 - 2Zz cos phi} = |vec{Z} - vec{z}|$.
Does anyone know of a generalization for Bessel functions of arbitrary order $J_m big(sqrt{Z^2+z^2 - 2Zz cos phi} big)$: $m$ is any integer?
summation bessel-functions
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
$endgroup$
$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
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In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33
add a comment |
$begingroup$
There is a Neumann theorem, proving that:
$J_0 big(sqrt{Z^2+z^2 - 2Zz cos phi} big)=sum_{k=0}^{infty} epsilon_k J_k(Z) J_k (z) cos k phi$
where $epsilon_0=1$, and $epsilon_k = 2$: $kgeq 1$. It is assumed that $sqrt{Z^2+z^2 - 2Zz cos phi} = |vec{Z} - vec{z}|$.
Does anyone know of a generalization for Bessel functions of arbitrary order $J_m big(sqrt{Z^2+z^2 - 2Zz cos phi} big)$: $m$ is any integer?
summation bessel-functions
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
$endgroup$
There is a Neumann theorem, proving that:
$J_0 big(sqrt{Z^2+z^2 - 2Zz cos phi} big)=sum_{k=0}^{infty} epsilon_k J_k(Z) J_k (z) cos k phi$
where $epsilon_0=1$, and $epsilon_k = 2$: $kgeq 1$. It is assumed that $sqrt{Z^2+z^2 - 2Zz cos phi} = |vec{Z} - vec{z}|$.
Does anyone know of a generalization for Bessel functions of arbitrary order $J_m big(sqrt{Z^2+z^2 - 2Zz cos phi} big)$: $m$ is any integer?
summation bessel-functions
summation bessel-functions
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
edited Dec 13 '18 at 21:21
MsTais
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
asked Dec 13 '18 at 20:30
MsTaisMsTais
1808
1808
$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
$begingroup$
In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33
add a comment |
$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
$begingroup$
In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33
$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
$begingroup$
In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33
add a comment |
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$begingroup$
The $x$ should be a $z$.
$endgroup$
– gammatester
Dec 13 '18 at 21:16
$begingroup$
In case one would need it, the generalization can be found here: quod.lib.umich.edu/u/umhistmath/ACV1415.0001.001/…
$endgroup$
– MsTais
Dec 13 '18 at 21:22
$begingroup$
There some more generalizations in Chap. 11, pp 358
$endgroup$
– gammatester
Dec 13 '18 at 21:33