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I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical soultion for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfacory in a certain sence. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

Attached sequence of numerical solutions of the truncated system

$\sum_{n=0}^{N}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

seems to indicate that the numerical solution slowly converges to an oscillating function.

Numerical solution for $N=60$ equations: $N=60$ equations" />

Numerical solution for $N=120$ equations: $N=120$ equations" />

Numerical solution for $n=240$ equations: $N=240$ equations" />

I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coefficients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical solution for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfactory in a certain sense. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

Attached sequence of numerical solutions of the truncated system

$\sum_{n=0}^{N}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

seems to indicate that the numerical solution slowly converges to an oscillating function.

Numerical solution for $N=60$ equations: $N=60$ equations" />

Numerical solution for $N=120$ equations: $N=120$ equations" />

Numerical solution for $n=240$ equations: $N=240$ equations" />

I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical soultion for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfacory in a certain sence. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

Attached series of numerical solutions below of the truncated system

$\sum_{n=0}^{N}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

seems to indicate that numerical solution slowly converges to an oscillating function.

Numerical solution for $N=60$ equations: $N=60$ equations" />

Numerical solution for $N=120$ equations: $N=120$ equations" />

Numerical solution for $n=240$ equations: $N=240$ equations" />

I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical soultion for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfacory in a certain sence. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

Attached sequence of numerical solutions of the truncated system

$\sum_{n=0}^{N}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

seems to indicate that the numerical solution slowly converges to an oscillating function.

Numerical solution for $N=60$ equations: $N=60$ equations" />

Numerical solution for $N=120$ equations: $N=120$ equations" />

Numerical solution for $n=240$ equations: $N=240$ equations" />

I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical soultion for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfacory in a certain sence. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

I have the series

$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$:

$a_0=1, a_{n}=2\times(-1)^n$.

It is related to a well known problem of Pierce's electrodes in physics. My problem also deals with the Pierce electrodes but in a different geometry. I found a numerical soultion for the most wanted value $\nu=2\pi/3$ using SVD pseudoinverse but is not satisfacory in a certain sence. Since the above equation for $a_{n}$ looks simple it is hoped that a simple solution to the problem might exist.

Attached series of numerical solutions below of the truncated system

$\sum_{n=0}^{N}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,

seems to indicate that numerical solution slowly converges to an oscillating function.

Numerical solution for $N=60$ equations: $N=60$ equations" />

Numerical solution for $N=120$ equations: $N=120$ equations" />

Numerical solution for $n=240$ equations: $N=240$ equations" />