Special Functions

This tutorial shows interfaces and examples of special_functions package.

Python Interface
Examples in Python
Cython Interface
Examples in Cython

Python Interface

Syntax	Return type	Symbol	Description
`besselj(nu, z, n)`	`double`, `double complex`	\[\partial^n J_{\nu}/\partial z^n\]	Bessel function of the first kind
`bessely(nu, z, n)`	`double`, `double complex`	\[\partial^n Y_{\nu}/\partial z^n\]	Bessel function of the second kind (Weber function)
`besseli(nu, z, n)`	`double`, `double complex`	\[\partial^n I_{\nu}/\partial z^n\]	Modified Bessel function of the first kind
`besselk(nu, z, n)`	`double`, `double complex`	\[\partial^n I_{\nu}/\partial z^n\]	Modified Bessel function of the second kind
`besselh(nu, k, z, n)`	`double complex`	\[\partial^n H^{(k)}_{\nu}/\partial z^n\]	Bessel function of the third kind (Hankel function)
`loggamma(nu, x, n)`	`double`	\[\log \Gamma(x)\]	Natural logarithm of Gamma function

Input Arguments

Variable	Type(s)	Symbol	Description
`nu`	`double`	\[\nu\]	The parameter of the Bessel functions.
`k`	`int`	\[k\]	Canbe either `1` and `2` and determines the type of Hankel function.
`z`	`double`, `double complex`	\[z\]	The real or complex input argument of the Bessel functions.
`x`	`double`	\[x\]	The real input argument of functions.

Examples in Python

In the example below, we compute the modified Bessel function of the second kind $ \partial`^n K_{:nbsphinx-math:nu`}(z) / \partial `z^n$ where :math:nu` is the parameter of the Bessel function, $z$ can be real or complex, and $n$ is the order of the derivative of the function. $n = 0$ means no derivative.

[3]:

import special_functions as sf
nu = 1.5
z = 2.0
n = 0
sf.besselk(nu, z, n)

[3]:

0.17990665795209218

Now, we try a complex argument $z = 2 + j$ (where $j^2 = -1$) and the first derivative of the function:

[12]:

z = 2.0 + 1.0j
n = 1
sf.besselk(nu, z, n)

[12]:

(-0.00937048840778705+0.21078407796464216j)

Cython Interface

In Cython, there are two syntaxes for each function depending on whether the input argument is real or complex.

For real arguments, the syntaxes are the same as the Python interface. For example: besselk.
For complex arguments, add the letter c to the begining of the function, for example: cbesselk.

The table below shows the function signatures for real and complex cases:

Real functions	Complex functions
`double besselj(double nu, double x, int n)`	`double complex cbesselj(double nu, double complex z, int n)`
`double bessely(double nu, double x, int n)`	`double complex cbessely(double nu, double complex z, int n)`
`double besseli(double nu, double x, int n)`	`double complex cbesseli(double nu, double complex z, int n)`
`double besselk(double nu, double x, int n)`	`double complex cbesselk(double nu, double complex z, int n)`
`double complex besselh(double nu, int k, double x, int n)`	`double complex cbesselh(double nu, int k, double complex z, int n)`
`double loggamma(double x)`	N/A

Examples in Cython

[19]:

%load_ext Cython

The Cython extension is already loaded. To reload it, use:
  %reload_ext Cython

[17]:

%%cython

# Use this in a *.pyx file
cimport special_functions as sf

# Declare typed input variables
cdef double nu = 1.5
cdef double x = 2.0
cdef int n = 0

# Declare typed output variable
cdef double K_real

# The 'nogil' statement below is optional.
with nogil:
    K_real = sf.besselk(nu, x, n)

# Print the output
print(K_real)

0.17990665795209218

Now, we try the same code in the above, but for the complex argument $z = 2.0 + j$.

[18]:

%%cython

# Use this in a *.pyx file
cimport special_functions as sf

# Declare typed input variables
cdef double nu = 1.5
cdef double complex z = 2.0 + 1.0j
cdef int n = 0

# Declare typed output variable
cdef double complex K_complex

# The 'nogil' statement below is optional.
with nogil:
    K_complex = sf.cbesselk(nu, z, n)

print(K_complex)

(0.031409508972862196-0.157309366284669j)