Bessel Function of the Third Kind

This module computes the Bessel function of the third kind (Hankel function) or its \(n\)^th derivative

\[\frac{\partial^n H^{(k)}_{\nu}(z)}{\partial z^n},\]

where

• \(n \in \mathbb{N}\) is the order of the derivative (\(n = 0\) indicates no derivative).

• \(k\) can be \(1\) or \(2\) and indicates the Hankel function of the first or second type, respectively.

• \(\nu \in \mathbb{R}\) is the order of the Bessel function.

• \(z \in \mathbb{C}\) is the input argument.

Syntax

This function has the following syntaxes depending on whether it is used in Python or Cython, or the input argument z is complex or real.

Interface	Input Type	Function Signature
Python	Real or Complex	besselh(nu, k, z, n=0)
Cython	Real	double complex besselh(double nu, int k, double z, int n)
	Complex	double complex cbesselh(double nu, int k, double complex z, int n)

Input Arguments:

nu: double

The parameter \(\nu\) of Bessel function.

k: int

It can be either 1 or 2 and determines the Hankel function of the first or second kind, respectively.

z: double or double complex

The input argument \(z\) of Bessel function.

• In Python, the function besselh accepts double and double complex types.

• In Cython, the function besselh accepts double type.

• In Cython, the function cbesselh accepts double complex type.

n: int = 0

The order \(n\) of the derivative of Bessel function. Zero indicates no derivative.

• For the Python interface, the default value is 0 and this argument may not be provided.

• For the Cython interfaces, no default value is defined and this argument should be provided.

See also

• Bessel function of the first kind.

• Bessel function of the second kind.

Examples

Using in Cython Code

The codes below should be used in a .pyx file and compiled with Cython.

As shown in the codes below, the python’s global lock interpreter, or gil, can be optionally released inside the scope of with nogil: statement. This is especially useful in parallel OpenMP environments.

Real Input

This example shows the real function besselh to compute the Bessel function of the third kind for a real argument z. The output variables d0h, d1h, and d2h are complex variables and represent the values of Bessel function and its first and second derivatives, respectively.

>>> # cimport module in a *.pyx file
>>> from special_functions cimport besselh

>>> # Declare typed variables
>>> cdef double nu = 2.5
>>> cdef int k = 1
>>> cdef double z = 2.0
>>> cdef double complex d0h, d1h, d2h

>>> # Releasing gil to secure maximum cythonic speedup
>>> with nogil:
...     d0h = besselh(nu, k, z, 0)    # no derivative
...     d1h = besselh(nu, k, z, 1)    # 1st derivative
...     d2h = besselh(nu, k, z, 2)    # 2nd derivative

Complex Input

The example below is similar to the above, except, the complex function cbesselh with complex argument z is used. The output variables d0h, d1h, and d2h are complex.

>>> # cimport module in a *.pyx file
>>> from special_functions cimport cbesselh

>>> # Declare typed variables
>>> cdef double nu = 2.5
>>> cdef int k = 1
>>> cdef double complex z = 2.0 + 1.0j
>>> cdef double complex d0h, d1h, d2h

>>> # Releasing gil to secure maximum cythonic speedup
>>> with nogil:
...     d0h = cbesselh(nu, k, z, 0)    # no derivative
...     d1h = cbesselh(nu, k, z, 1)    # 1st derivative
...     d2h = cbesselh(nu, k, z, 2)    # 2nd derivative

Using in Python Code

The codes below should be used in a .py file and no compilation is required. The python’s global lock interpreter, or gil, cannot be released.

Real Input

The example below uses the function besselh with the real argument z to compute the Bessel function of the third kind and its first and second derivatives.

>>> # import module in a *.py file
>>> from special_functions import besselh

>>> nu = 2.5
>>> k = 1
>>> z = 2.0

>>> d0h = besselh(nu, k, z)       # no derivative
>>> d1h = besselh(nu, k, z, 1)    # 1st derivative
>>> d2h = besselh(nu, k, z, 2)    # 2nd derivative

Complex Input

To use a complex input argument z in the Python interface, the same function besselh as the previous example can be used. This is unlike the Cython interface in which cbesselh should be used.

>>> # import module in a *.py file
>>> from special_functions import besselh

>>> nu = 2.5
>>> k = 1
>>> z = 2.0 + 1.0j

>>> d0h = besselh(nu, k, z)       # no derivative
>>> d1h = besselh(nu, k, z, 1)    # 1st derivative
>>> d2h = besselh(nu, k, z, 2)    # 2nd derivative

Tests

The test script of this module is located at tests/test_besselh.py. The test compares the results of this module with scipy.special package (functions j0, j1, jn, jv, and jvp) for several combinations of input parameters with multiple values. Run the test by

git clone https://github.com/ameli/special_functions.git
cd special_functions/tests
python test_besselh.py

Algorithm

Depending on the values of the input parameters \((\nu, z, n)\), one of the following two algorithms is employed.

• If \(\nu + \frac{1}{2} \in \mathbb{Z}\), the Bessel function is computed using half-integer formulas in terms of elementary functions.

• For other cases, the computation is carried out by Amos Fortran library (see [Amos-1986]) using zbesh subroutine in that library.

Special Cases

In the special cases below, the computation is performed by taking advantage of some of the known formulas and properties of the Bessel functions.

Negative \(\nu\)

When \(\nu < 0\) and for the two cases below, the Bessel function is related to the Bessel function of the positive parameter \(-\nu\).

• If \(\nu \in \mathbb{Z}\) (see [DLMF] Eq. 10.4.1):

  \[H^{(k)}_{\nu}(z) = (-1)^{\nu} H^{(k)}_{-\nu}(z),\]

  where \(k = 1, 2\).

• If \(\nu + \frac{1}{2} \in \mathbb{Z}\) (see [DLMF] Eq. 10.2.3):

  \[H^{(k)}_{\nu}(z) = \left( \cos(\pi \nu) - i \alpha(k) \sin(\pi \nu) \right) H^{(k)}_{-\nu}(z),\]

  where \(k = 1, 2\), \(\alpha(1) = 1\), and \(\alpha(2) = -1\).

Derivatives

If \(n > 0\), the following relation for the derivative is applied (see [DLMF] Eq. 10.6.7):

\[\frac{\partial^n H^{(k)}_{\nu}(z)}{\partial z^n} = \frac{1}{2^n} \sum_{i = 0}^n (-1)^i \binom{n}{i} H^{(k)}_{\nu - n + 2i}(z),\]

where \(k = 1, 2\).

Half-Integer \(\nu\)

When \(\nu\) is half-integer, the Bessel function is computed in terms of elementary functions as follows.

• If \(z = 0\), then NAN is returned.

• If \(z < 0\) and \(z \in \mathbb{R}\), then NAN is returned.

• If \(\nu = \pm \frac{1}{2}\) (see [DLMF] Eq. 10.16.1)

  \[\begin{split}H^{(k)}_{\frac{1}{2}}(z) = \sqrt{\frac{2}{\pi z}} \left( \sin(z) - i \alpha(k) \cos(z) \right), \\ H^{(k)}_{-\frac{1}{2}}(z) = \sqrt{\frac{2}{\pi z}} \left( \cos(z) + i \alpha(k) \sin(z) \right),\end{split}\]

  where \(k = 1, 2\) and \(\alpha(1) = 1\) and \(\alpha(2) = -1\). Depending on \(z\), the above relations are computed using the real or complex implementation of the elementary functions.

• Higher-order half-integer parameter \(\nu\) is related to the above relation for \(\nu = \pm \frac{1}{2}\) using recursive formulas (see [DLMF] Eq. 10.6.1):

\[\begin{split}H^{(k)}_{\nu}(z) = \frac{2 (\nu - 1)}{z} H^{(k)}_{\nu - 1}(z) - H^{(k)}_{\nu - 2}(z), \qquad \nu > 0, \\ H^{(k)}_{\nu}(z) = \frac{2 (\nu + 1)}{z} H^{(k)}_{\nu + 1}(z) - H^{(k)}_{\nu + 2}(z), \qquad \nu < 0,\end{split}\]

where \(k = 1, 2\).

References

[Cephes-1989]

Moshier, S. L. (1989). C language library with special functions for mathematical physics. Available at http://www.netlib.org/cephes.

[Amos-1986]

Amos, D. E. (1986). Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12, 3 (Sept. 1986), 265-273. DOI: https://doi.org/10.1145/7921.214331. Available at http://netlib.org/amos/.

[DLMF] (1,2,3,4,5)

Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S., and McClain, M. A., eds. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.0 of 2020-12-15.