I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is fine until I get till higher $f$ coefficients.
$J_0$ is the Bessel function.
I've evaluated the function for a couple of values of $f$, and at $f=49.46$, you already see the errors arising.
Note: the graphs have been divided by half the value at $\cos{\gamma}$, such that the graphs coincide at 0.5.
Apparently the solution is to expand the integral in in spherical harmonics. But since the direct numerical projection of spherical harmonics leads to large errors because numerical integration of rapidly oscillating functions is inaccurate, I need to use an interpolating basis. But I consider that as a problem for later.
If you want more background information, I can give it.