Modified Bessel Function of the Second Kind

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

\[\frac{\partial^n K_{\nu}(z)}{\partial z^n},\]

where

\(n \in \mathbb{N}\) is the order of the derivative (\(n = 0\) indicates no derivative).
\(\nu \in \mathbb{R}\) is the order of the modified 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	`besselk(nu, z, n=0)`
Cython	Real	`double besselk(double nu, double z, int n)`
Cython	Complex	`double complex cbesselk(double nu, double complex z, int n)`

Input Arguments:

nu: double: The parameter \(\nu\) of modified Bessel function.

z: double or double complex

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

In Python, the function besselk accepts double and double complex types.
In Cython, the function besselk accepts double type.
In Cython, the function cbesselk accepts double complex type.

n: int = 0

The order \(n\) of the derivative of modified 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.

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 Function

This example shows the real function besselk to compute the modified Bessel function of the second kind for a real argument z. The output variables d0k, d1k, and d2k represent the values of modified Bessel function and its first and second derivatives, respectively.

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

>>> # Declare typed variables
>>> cdef double nu = 2.5
>>> cdef double z = 2.0
>>> cdef double d0k, d1k, d2k

>>> # Releasing gil to secure maximum cythonic speedup
>>> with nogil:
...     d0k = besselk(nu, z, 0)    # no derivative
...     d1k = besselk(nu, z, 1)    # 1st derivative
...     d2k = besselk(nu, z, 2)    # 2nd derivative

Complex Function

The example below is similar to the above, except, the complex function cbesselk with complex argument z is used. The output variables d0k, d1k, and d2k are also complex.

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

>>> # Declare typed variables
>>> cdef double nu = 2.5
>>> cdef double complex z = 2.0 + 1.0j
>>> cdef double complex d0k, d1k, d2k

>>> # Releasing gil to secure maximum cythonic speedup
>>> with nogil:
...     d0k = cbesselk(nu, z, 0)    # no derivative
...     d1k = cbesselk(nu, z, 1)    # 1st derivative
...     d2k = cbesselk(nu, 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 Function

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

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

>>> nu = 2.5
>>> z = 2.0

>>> d0k = besselk(nu, z)       # no derivative
>>> d1k = besselk(nu, z, 1)    # 1st derivative
>>> d2k = besselk(nu, z, 2)    # 2nd derivative

Complex Function

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

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

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

>>> d0k = besselk(nu, z)       # no derivative
>>> d1k = besselk(nu, z, 1)    # 1st derivative
>>> d2k = besselk(nu, z, 2)    # 2nd derivative

Tests

The test script of this module is located at tests/test_besselk.py. The test compares the results of this module with scipy.special package (functions k0, k1, kn, kv, and kvp) 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_besselk.py

Algorithm

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

If \(z \in \mathbb{R}\) (that is, z is of type double) and \(\nu = 0\) or \(\nu = 1\), the computation is carried out by Cephes C library (see [Cephes-1989]), respectively using k0, k1, or kn functions in that library.
If \(\nu + \frac{1}{2} \in \mathbb{Z}\), the modified 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 zbesk 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 modified Bessel functions.

Branch-Cut

In the real domain where \(z \in\mathbb{R}\), if \(z < 0\), the value of NAN is returned.
However, in the complex domain \(z \in\mathbb{C}\) and on the branch-cut of the function, \(\Re(z) < 0\) and \(\Im(z) = 0\), its principal value corresponding to the branch

\[\mathrm{arg}(z) \in (-\pi, \pi]\]

is returned. This value may be finite, infinity or undefined depending on \(\nu\) and \(z\).

Negative \(\nu\)

When \(\nu < 0\) the modified Bessel function is computed is related to the modified Bessel function of the positive parameter \(-\nu\) by (see [DLMF] Eq. 10.27.3):

\[K_{\nu}(z) = K_{-\nu}(z).\]

Derivatives

If \(n > 0\), the following relation for the derivative is applied (see [DLMF] Eq. 10.29.5):

\[\frac{\partial^n K_{\nu}(z)}{\partial z^n} = \frac{1}{2^n} \sum_{i = 0}^n \binom{n}{i} K_{\nu - n + 2i}(z).\]

Half-Integer \(\nu\)

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

If \(z = 0\):
- If \(z \in \mathbb{R}\), then \(K_{\nu}(0) = +\infty\).
- If \(z \in \mathbb{C}\), then NAN is returned.
If \(\nu = \pm \frac{1}{2}\) (see [DLMF] Eq. 10.39.2)

\[K_{\pm\frac{1}{2}}(z) = \sqrt{\frac{\pi}{2 z}} \exp(-z).\]

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):

\[K_{\nu}(z) = \frac{2 (|\nu| - 1)}{z} K_{|\nu| - 1}(z) + K_{|\nu| - 2}(z).\]

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)

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.