A Big Table of Equations
A reference of useful functions/tables for learning & solving differential equations
At some point it gets difficult to memorize and reference these particular functions or tables. So to make my life easier I have decided to undertake the challenge of collecting and organizing this information all into one place.
I plan to come back often and update this page, so if there are any suggestions or changes to make to make it easier for others to reference as well, please feel free to do so.
Orthogonal Functions
Hermite Polynomials \( H_{n} \)
Formula: \[H_n(x) = (-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}} \]
Description:
Part of the solution to the Quantum Harmonic Oscillator
Associated Laguerre Polynomials \( L_{n}^{k} \)
Differential Equation: \[ xy''+ (k+1-x)y'+ny = 0 \]
Formula: \[ L_{n}^{k}(x)=\sum^{n}_{m=0}\frac{(n-k)!}{(n-m)!(k+m)!}\frac{x^{k}}{m!} \]
Description:
Used in the bound state solution for the radial component of the hydrogen atom
Associated Legendre Polynomials \( P_{l}^{m} \)
Differential Equation: \[ (1-x^{2})y'' -2xy' + \left[l(l+1)-\frac{m^{2}}{1-x^{2}} \right]ny = 0 \]
Formula: \[ P_{l}^{m}(x)=\frac{(-1)^{m}}{2^{l}l!}(1-x^{2})^{m/2}\frac{d^{l+m}}{dx^{l+m}} \]
Description:
Used in the angular solution to the time-indpendent schrodinger equation for any radial potential
*not reliable* Spherical Bessel Function \( j_{l}^{x} \)
Differential Equation: \[ (1-x^{2})y'' -2xy' + \left[l(l+1)-\frac{m^{2}}{1-x^{2}} \right]ny = 0 \]
Formula: \[ P_{l}^{m}(x)=\frac{(-1)^{m}}{2^{l}l!}(1-x^{2})^{m/2}\frac{d^{l+m}}{dx^{l+m}} \]
Description:
Used in the angular solution to the time-indpendent schrodinger equation for any radial potential
*not reliable*Spherical Neumann Function \( y_{l}^{x} \)
Differential Equation: \[ (1-x^{2})y'' -2xy' + \left[l(l+1)-\frac{m^{2}}{1-x^{2}} \right]ny = 0 \]
Formula: \[ P_{l}^{m}(x)=\frac{(-1)^{m}}{2^{l}l!}(1-x^{2})^{m/2}\frac{d^{l+m}}{dx^{l+m}} \]
Description:
Used in the angular solution to the time-indpendent schrodinger equation for any radial potential