¿Could somebody tell me how can i write a zero order bessel function in an Hermite-Gauss basis? Thanks
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The Hermite-Gauss functions (suitably normalized) are orthonormal for an inner product defined by integration, just like ${\sin(nx),\cos(mx)}$ as $n,m$ run over positive integers. The coefficients of an expansion of a $J$-Bessel function in terms of Hermite-Gauss functions are computed by integrating the Bessel functions against the various Hermite-Gauss functions, in a way analogous to how a Fourier expansion is computed. You'll want to be looking at a table of integrals, or maybe Mathematica. (I'm not guaranteeing anything about the convergence of the series; I haven't thought about it that much.) |
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Thank you very much! I was thinking in an expansion tabulated yet like those appearing in Gradstein and Ryzhik, but i didn't find it anywhere. |
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