As part of my final year project, I'm using the identity where J is a Bessel function of the first kind.
Instead of just quoting this, it would be quite nice to actually prove it, so I was wondering if anyone could direct me in how to do so?
As part of my final year project, I'm using the identity where J is a Bessel function of the first kind.
Instead of just quoting this, it would be quite nice to actually prove it, so I was wondering if anyone could direct me in how to do so?
I'm using this definition, which is probably yours (?): .
Note that is the -th complex Fourier coefficient of the function (extended periodically). The general theory says that its Fourier series converges: for all . Take the real part of both sides, and notice that so that terms with odd indices cancel, while terms with even (non-zero) indices count twice (this gives the 2 in your formula). That's it.