There is a classic Hankel transform pair from optics that I am trying to reproduce with Mathematica, without luck, so I can confidently generate an unknown case. The function in the spatial frequency domain, H, is:
H[rho_]:=2/Pi*(ArcCos[rho]-rho*Sqrt[1-rho^2])
with rho < 1. There are several versions of the normalized inverse Hankel transform of the frequency domain function available in the literature with hNorm[r] = h[r]/h[0] where h is the inverse hankel transform of H. I'd be happy to find any of them. For discussion sake one is published to be:
hNorm[r_]:=BesselJ[1,r/2]^2/(r/2)^2
When I try the inversion, however, I only get a result for the rho*Sqrt[1-rho^2] term. The inverse transform for the ArcCos term (hStar) is returned unchanged:
hStar[r_]=Assuming[rho<1,InverseHankelTransform[ArcCos[rho],rho,r]]
yields,
InverseHankelTransform[ArcCos[rho],rho,r,0]
I've tried to enforce various appropriate assumptions and have tried other Bessel function orders in the transform without luck. Any ideas?
Thanks