Fourier transform of Bessel functions

[vietha]

Hi there,

I am calculating the Fourier transform of the bessel function $$J_0^2(bx)$$ by using Maple. I tried two equations and get two results.

$$\int J_0^2(bx)e^{-j2\pi fx}dx=G^{2, 1}_{2, 2}\left(-1/4\,{\frac {{w}^{2}}{{b}^{2}}}\, \Big\vert\,^{1/2, 1/2}_{0, 0}\right) {\pi }^{-1}{b}^{-1}$$

and

$$\int J_0^2(bx)[cos(2\pi fx)-jsin(2\pi fx)]dx=G^{2, 0}_{2, 2}\left(1/4\,{\frac {{w}^{2}}{{b}^{2}}}\, \Big\vert\,^{1/2, 1/2}_{0, 0}\right) {b}^{-1}$$

I tried to plot these two functions but only the second one shows a plot. The first one does not show anything and it tells that could not evaluate the function in the variable range (e.g. 0..4).

Could anyone help me to verify this? I want to make sure that the two results are identical. If it is not, which one is correct so I may continue with other calculations of FFT on the bessel functions and on the Generalized geometric functions.


Thanks so much in advance.

VietHa

 

[jvicens]

Dear VietHa,

Thank you for sharing your calculations with us. It is always important to verify the results of your calculations, especially when dealing with complex functions like the Fourier transform of the bessel function. I have taken a look at the two equations you provided and have some insights to share with you.

Firstly, both equations are correct, but they are representing different forms of the same function. The first equation is using the generalized geometric function, while the second equation is using the cosine and sine functions. The reason why the first equation is not showing a plot is because it is undefined for certain values of the variable range, as indicated by the error message you received. This is because the generalized geometric function has a specific range of convergence, and if the values fall outside of that range, it cannot be evaluated.

To verify the equivalence of the two equations, you can use the properties of the generalized geometric function to simplify the first equation and obtain the second equation. This will show that they are indeed the same function, just represented differently.

In terms of which one to use for your calculations, it ultimately depends on your specific needs and the range of values you are working with. If your values fall within the range of convergence for the generalized geometric function, then it would be more accurate to use the first equation. However, if your values fall outside of that range, then the second equation would be a better choice.

I hope this helps to clarify the situation for you. Good luck with your further calculations and experiments with the Fourier transform of the bessel function.