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## Summary:

- I am trying to formulate an analytical expression for the spectrum of the electric fields on a circular aperture (cylindrical waveguide). The field expressions are a multiplication of Bessel function and sinusoidal function. I am attaching only one kind of integration that I have.

## Main Question or Discussion Point

[tex] I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi [/tex]

Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as [itex] \int_{0}^{2\pi} \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} d\phi [/itex] but then again I am not able to find any solution.

Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as [itex] \int_{0}^{2\pi} \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} d\phi [/itex] but then again I am not able to find any solution.