Install from PyPi

The recommended installation method is through the package available at PyPi using pip.

1. Ensure pip is installed within Python and upgrade the existing pip by

   python -m ensurepip
   python -m pip install --upgrade pip

   If you are using PyPy instead of Python, ensure pip is installed and upgrade the existing pip by

   pypy -m ensurepip
   pypy -m pip install --upgrade pip

2. Install this package in Python by

   python -m pip install special_functions

   or, in PyPy by

   pypy -m pip install special_functions

Install from Anaconda Cloud

Alternatively, the package can be installed through Anaconda could.

• In Linux and Windows:

  conda install -c s-ameli special_functions

• In macOS:

  conda install -c s-ameli -c conda-forge special_functions

Build and Install from Source Code

Build dependencies: To build the package from the source code, numpy and cython are required. These dependencies are installed automatically during the build process and no action is needed.

1. Install both C and Fortran compilers as follows.

   • Linux: Install gcc, for instance, by apt (or any other package manager on your Linux distro)

     sudo apt install gcc gfortran

   • macOS: Install gcc via Homebrew:

     sudo brew install gcc

     Note: If gcc is already installed, but Fortran compiler is yet not available on macOS, you may resolve this issue by reinstalling:

     sudo brew reinstall gcc

   • Windows: Install both Microsoft Visual C++ compiler and Intel Fortran compiler (Intel oneAPI). Open the command prompt (where you will enter the installation commands in the next step) and load the Intel compiler variables by

     C:\Program Files (x86)\Intel\oneAPI\setvars.bat

     Here, we assumed the Intel Fortran compiler is installed in C:\Program Files (x86)\Intel\oneAPI. You may set this directory accordingly to the directory of your Intel compiler.

2. Clone the source code and install this package by

   git clone https://github.com/ameli/special_functions.git
   cd special_functions
   python -m pip install .

Warning: After the package is built and installed from the source code, the package cannot be imported properly if the current working directory is the same as the source code directory. To properly import the package, change the current working directory to a directory anywhere else outside of the source code directory. For instance:

cd ..
python
>>> import special_functions

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

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