Bessel transforms

Let $f(x)$ a real function defined on an interval $[a,b]$, $n\geq 0$ an integer. The $n$-th Bessel transform of $f$ is the function

$$B_nf(t) \ = \ \int_a^bj_n(tx)f(x)dx,$$

where $j_n$ is the Bessel function of order $n$. Let $Tf(t)$ the Fourier transform of $f(x)$ and $Rf(t)$, $If(t)$ the real and imaginary part of $Tf(t)$:

$$Rf(t) \ = \ \int_a^b\cos(tx)f(x)dx, \ \ \ \ If(t) \ = \ \int_a^b\sin(tx)f(x)dx.$$

Then we have

$$B_nf(t) \ = \ \frac{2}{\pi}\int_0^{\pi/2}Rf(t\sin\theta)\cos(n\theta)d\theta \ \ \ \ {\rm if \ }n \ {\rm is \ even},$$

$$B_nf(t) \ = \ \frac{2}{\pi}\int_0^{\pi/2}If(t\sin\theta)\sin(n\theta)d\theta \ \ \ \ {\rm if \ }n \ {\rm is \ odd}.$$

We use these formulas to compute Bessel transforms in funct.