acoular.sources.spherical_hn1¶

acoular.sources.spherical_hn1(n, z)¶

Compute the spherical Hankel function of the first kind.

The spherical Hankel function of the first kind, \(h_n^{(1)}(z)\), is defined as

\[h_n^{(1)}(z) = j_n(z) + i \cdot y_n(z)\]

with the complex unit \(i\), the spherical Bessel function of the first kind as

\[j_n(z) = \sqrt{\frac{\pi}{2z}} J_{n + 1/2}(z),\]

and the spherical Bessel function of the second kind as

\[y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),\]

where \(Y_n\) is the Bessel function of the second kind.

Parameters:

nint, array_like

Order of the spherical Hankel function. Must be a non-negative integer.

zcomplex or float, array_like

Argument of the spherical Hankel function. Can be real or complex.

Returns:

complex or numpy.ndarray

Value of the spherical Hankel function of the first kind for the given order n and argument z. If z is array-like, an array of the same shape is returned.

See also

scipy.special.spherical_jn()

Computes the spherical Bessel function of the first kind.

scipy.special.spherical_yn()

Computes the spherical Bessel function of the second kind.

Notes

• The function relies on scipy.special.spherical_jn() for the spherical Bessel function of the first kind and scipy.special.spherical_yn() for the spherical Bessel function of the second kind.

• The input n must be a non-negative integer; otherwise, the behavior is undefined.

Examples

>>> import acoular as ac
>>>
>>> ac.sources.spherical_hn1(0, 1.0)
np.complex128(0.8414709848078965-0.5403023058681398j)
>>> ac.sources.spherical_hn1(1, [1.0, 2.0])
array([0.30116868-1.38177329j, 0.43539777-0.350612j  ])