scipy.special.hankel1#

scipy.special.hankel1(v, z, out=None) = <ufunc 'hankel1'>#

Hankel function of the first kind

Parameters:

varray_like: Order (float).
zarray_like: Argument (float or complex).
outndarray, optional: Optional output array for the function values

Returns:

scalar or ndarray: Values of the Hankel function of the first kind.

See also

hankel1e: This function with leading exponential behavior stripped off.

Notes

A wrapper for the AMOS [1] routine zbesh, which carries out the computation using the relation,

\[H^{(1)}_v(z) = \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2))\]

where \(K_v\) is the modified Bessel function of the second kind. For negative orders, the relation

\[H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v)\]

is used.

References

[1]

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/

Examples

For the inhomogenous Helmholtz equation in \(\mathbb{R}^2\) subject to radiation boundary conditions, the Green’s function is given by

\[G(\mathbf{x}, \mathbf{x}^\prime) = \frac{1}{4i} H^{(1)}_0(k|\mathbf{x} - \mathbf{x^\prime}|)\]

where \(k\) is the wavenumber and \(H^{(1)}_0\) is the Hankel function of the first kind and of order zero. In the following example, we will solve the Helmholtz equation with two Dirac sources.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import hankel1
>>> k = 10  # Wavenumber
>>> sources = [0.5, -0.5]  # Location of two point sources
>>> x, y = np.linspace(-3, 3, 300), np.linspace(-3, 3, 300)
>>> Z = np.add.outer(1j*y, x)
>>> U = np.zeros_like(Z)
>>> for sz in sources:
...     r = np.abs(Z - sz)
...     U += (1j/4)*hankel1(0, k*r)

Finally, we will plot the real part of the solution.

>>> fig, ax = plt.subplots()
>>> ax.pcolormesh(np.real(Z), np.imag(Z), np.real(U))
>>> plt.show()

../../_images/scipy-special-hankel1-1.png