Simple example of PyHank usage

In this example (as in the Example of single-shot transform) we will check the band limit of a jinc function: \(f(r) = \frac{J_1(r)}{r}\). The (0 order) Hankel transform of this should be the top hat function.

Here we create a HankelTransform object and use its qdht() method. In this simple case, the simpler, single shot functions used in Example of single-shot transform may be simpler to use. It should be noted, however, that they are not well suited for multiple transforms on the same grid and the approach taken here is recommended.

First import the HankelTransform class and other packages

from pyhank import HankelTransform
import scipy.special
import matplotlib.pyplot as plt

Create a HankelTransform object which holds the grid for \(r\) and \(k_r\) points and calculate the jinc function.

Note that although the calculation fails at \(r = 0\), transformer.r does not include \(r=0\).

transformer = HankelTransform(order=0, max_radius=100, n_points=1024)
f = scipy.special.jv(1, transformer.r) / transformer.r

plt.figure()
plt.plot(transformer.r, f)
plt.xlabel('Radius /m')

Now take the Hankel transform using HankelTransform.qdht()

ht = transformer.qdht(f)

plt.figure()
plt.plot(transformer.kr, ht)
plt.xlim([0, 5])
plt.xlabel('Radial wavevector /m$^{-1}$')

As expected, this is a top-hat function bandlimited to \(k<1\), except for numerical error.