.. _mathfunc:

Special Mathematical Functions
++++++++++++++++++++++++++++++

Hermite polynomials
-------------------

.. note::

   * What? Solution of the 1D Harmonic Oscilator
   * Where? :ref:`chapter7`
   * Wave-function: :math:`\psi(y)=h(y) e^{-y^{2} / 2}` where :math:`h(y)=\sum_{k=0}^{k_{max}} a_{k} y^{k}`
   * Recursion relation: :math:`\frac{a_{k+2}}{a_{k}}=\frac{2 k+1-\varepsilon}{(k+2)(k+1)}` (:math:`a_{k+2}=0 \text { for } k\ge n=\frac{\varepsilon-1}{2}`)

.. image:: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Hermite_poly_phys.svg/1920px-Hermite_poly_phys.svg.png
   :width: 600
   :alt: Hermite polynomials
   :align: center	 

Spherical harmonics
-------------------

.. note::

   * What? Solution of the angular part of the Schroedinger equation of a central potential
   * Where? :ref:`chapter9`
   * Wave-function: :math:`Y_{l, m}(\theta, \phi)=\frac{(-1)^{l}}{2^{l} l !} \sqrt{\frac{(2 l+1)(l+m) !}{4 \pi(l-m) !}} e^{i m \phi} \frac{1}{\sin ^{m} \theta} \frac{d^{l-m}}{d(\cos \theta)^{l-m}} \sin ^{2 l} \theta` 
   * First few soutions:

.. math::

   \begin{aligned}
   Y_{0,0}(\theta, \phi) &=\sqrt{\frac{1}{4 \pi}} \\
   Y_{1,\pm 1}(\theta, \phi) &=\mp \sqrt{\frac{3}{8 \pi}} e^{\pm i \phi} \sin \theta \\
   Y_{1,0}(\theta, \phi) &=\sqrt{\frac{3}{4 \pi}} \cos \theta \\
   Y_{2,\pm 2}(\theta, \phi) &=\sqrt{\frac{15}{32 \pi}} e^{\pm 2 i \phi} \sin ^{2} \theta \\
   Y_{2,\pm 1}(\theta, \phi) &=\mp \sqrt{\frac{15}{8 \pi}} e^{\pm i \phi} \sin \theta \cos \theta \\
   Y_{2,0}(\theta, \phi) &=\sqrt{\frac{5}{16 \pi}}\left(3 \cos ^{2} \theta-1\right)
   \end{aligned}

.. image:: https://i1.wp.com/opticaltweezers.org/wp-content/uploads/2015/11/Fig5_2.png?w=1320
   :width: 600
   :alt: Spherical harmonics
   :align: center

	
Legendre Polynomial
-------------------

Laguerre Polynomial
-------------------

Spherical Bessel Functions
--------------------------

.. math::

   \begin{array}{l}
   j_{0}(\rho)=\frac{\sin \rho}{\rho} \\
   j_{1}(\rho)=\frac{\sin \rho}{\rho^{2}}-\frac{\cos \rho}{\rho} \\
   j_{2}(\rho)=\left(\frac{3}{\rho^{3}}-\frac{1}{\rho}\right) \sin \rho-\frac{3 \cos \rho}{\rho^{2}}
   \end{array}

   
Spherical Neumann Functions
---------------------------

.. math::

   \begin{array}{l}
   \eta_{0}(\rho)=-\frac{\cos \rho}{\rho} \\
   \eta_{1}(\rho)=-\frac{\cos \rho}{\rho^{2}}-\frac{\sin \rho}{\rho} \\
   \eta_{2}(\rho)=-\left(\frac{3}{\rho^{3}}-\frac{1}{\rho}\right) \cos \rho-\frac{3 \sin \rho}{\rho^{2}}
   \end{array}
   
https://dlmf.nist.gov