fftl.transforms.hankel

fftl.transforms.hankel(mu, r, ar, *args, **kwargs)

Hankel transform

The Hankel transform is here defined as

\[\tilde{a}(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, r \, dr \;,\]

where \(J_\mu\) is the Bessel function of order \(\mu\). The order can in general be any real or complex number. The transform is orthogonal and normalised: applied twice, the original function is returned.

The Hankel transform is equivalent to a normalised spherical Hankel transform (sph_hankel()) with the order and bias shifted by one half. Special cases are \(\mu = 1/2\), which is related to the Fourier sine transform,

\[\tilde{a}(k) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} \! a(r) \, \frac{\sin(kr)}{\sqrt{kr}} \, r \, dr \;,\]

and \(\mu = -1/2\), which is related to the Fourier cosine transform,

\[\tilde{a}(k) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} \! a(r) \, \frac{\cos(kr)}{\sqrt{kr}} \, r \, dr \;.\]

Examples

Compute the Hankel transform for parameter mu = 1.

>>> # some test function
>>> p, q = 2.0, 0.5
>>> r = np.logspace(-2, 2, 1000)
>>> ar = r**p*np.exp(-q*r)
>>>
>>> # compute a biased transform
>>> from fftl.transforms import hankel
>>> mu = 1.0
>>> k, ak = hankel(mu, r, ar, q=0.1)

Compare with the analytical result.

>>> from scipy.special import gamma, hyp2f1
>>> res = k*q**(-3-p)*gamma(3+p)*hyp2f1((3+p)/2, (4+p)/2, 2, -(k/q)**2)/2
>>>
>>> import matplotlib.pyplot as plt
>>> plt.plot(k, ak, '-k', label='numerical')
>>> plt.plot(k, res, ':r', label='analytical')
>>> plt.xscale('log')
>>> plt.yscale('symlog', linthresh=1e0, subs=[2, 3, 4, 5, 6, 7, 8, 9])
>>> plt.ylim(-5e-1, 1e2)
>>> plt.legend()
>>> plt.show()