MATH3271

Formula Sheet

Fourier series

If f(x) is a 2p-periodic peicewise smooth function then its Fourier series is given by

where

Bessel functions

where λ_mn = α_mn/a and α_mn is the n^th positive zero of J_m(x).

Spherical Bessel functions

where λ_mj = α_m+1/2,j/a.

Legendre polynomials

Spherical harmonics

Curvilinear coordinates

Given a set of orthogonal curvilinear coordinates (u_1, u₂, u₃) we have the following expressions:

where f is a scalar function of (u_1, u₂, u₃) while F = F₁u₁ + F₂u₂ + F₃u₃ is a vector function of

(u_1, u₂, u₃).

For the cylindrical polar coordinate system

For the spherical polar coordinate system