hankel.hankel.HankelTransform.transform

HankelTransform.transform(f, k=1, ret_err=True, ret_cumsum=False, inverse=False)

Do the Hankel-transform of the function f.

Parameters:

f : callable

A function of one variable, representing \(f(x)\)

ret_err : boolean, optional, default = True

Whether to return the estimated error

ret_cumsum : boolean, optional, default = False

Whether to return the cumulative sum

Returns:

ret : array-like

The Hankel-transform of f(x) at the provided k. If k is scalar, then this will be scalar.

err : array-like

The estimated error of the approximate integral, at every k. It is merely the last term in the sum. Only returned if ret_err=True.

cumsum : array-like

The total cumulative sum, for which the last term is itself the transform. One can use this to check whether the integral is converging. Only returned if ret_cumsum=True

Notes

The Hankel transform is defined as

\[F(k) = \int_0^\infty r f(r) J_\nu(kr) dr.\]

The inverse transform is identical (swapping k and r of course).