xopto.util.hankel module¶

continuous(frequency: numpy.ndarray, rfun: numpy.ndarray, rstop: float, out: Optional[numpy.ndarray] = None, **kwargs) → numpy.ndarray[source]¶

Computes Hankel transform of a continuous radially symmetric function:

$g(q) &= 2 \pi \int_0^{\infty}f(r)J_0(2 \pi q r) r dr$

Parameters

frequency (np.ndarray) – A list of frequencies (1/m) at which to compute the Hankel transform.
rfun (callable) – Callable with one parameter (radius) representing a radially symmetric function.
rstop (float) – Range of numerical integration as [0, rstop].
out (np.ndarray) – Optional output array for the computed frequencies.
kwargs (dict) – Optional keyword arguments passed to the scipy.integrate.quad() function.

Returns

F – The Hankel transfor of rfun at the given frequencies.

Return type

np.ndarray vector

discrete(frequency: numpy.ndarray, rpts: numpy.ndarray, fpts: numpy.ndarray, logscale: bool = True, **kwargs) → numpy.ndarray[source]¶

Computes Hankel transform of a discrete function. The discrete function is first made continuous by means of interpolation in linear or log scale. Finally, the transform is computed by the continuous() function using quad.

$g(q) &= 2 \pi \int_0^{\infty}f(r)J_0(2 \pi q r) r dr$

Parameters

frequency (np.ndarray) – A list of frequencies (1/m) at which to compute the Hankel transform.
rpts (np.ndarray) – Points at which the discrete radially symmetric function is defined.
fpts (np.ndarray) – Value of the radially symmetric function at points rpts.
kwargs (dict) – Optional keyword arguments passed to the scipy.integrate.quad() function.

Returns

F – The Hankel transfor of rfun at the given frequencies.

Return type

np.ndarray

discrete_simpson(frequency: numpy.ndarray, rpts: numpy.ndarray, fpts: numpy.ndarray, uneven: Optional[bool] = None) → numpy.ndarray[source]¶

Computes Hankel transform of a radially symmetric function defined on a grid of evenly or unevenly spaced points. To compute transforms of multiple sets (functions), the fpts array shape must be (num_sets, rpts.size).

$g(q) &= 2 \pi \int_0^{\infty}f(r)J_0(2 \pi q r) r dr$

Parameters

frequency (np.ndarray) – A list of frequencies (1/m) at which to compute the Hankel transform.
rpts (np.ndarray) – A vector of evenly or unevenly spaced points at which the function values in fpts are defined.
fpts (np.ndarray) – A vector or array of function values defined at points rpts. To compute transforms of multiple sets (functions), the fpts array shape must be (num_sets, rpts.size).
uneven (bool) – If True, the method assumes unevenly spaced values in rpts. Default is False. If set to None, the value of uneven flag is derived from the values in the rpts array.

Returns

F – The Hankel transfor of rfun at the given frequencies. If the fpts array ia a vector (points of one function only) then F is a vector of size len(frequencies). If fpts is a 2D array of shape (N, rpts.size) then F is a 2D array of shape (N, len(frequencies)).

Return type

np.ndarray