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 λmn = αmn/a and αmn is the nth positive zero of Jm(x).

 

Spherical Bessel functions

where λmj = αm+1/2,j/a.

 

Legendre polynomials


Spherical harmonics

 

Curvilinear coordinates

 

            Given a set of orthogonal curvilinear coordinates (u1, u2, u3) we have the following expressions:

where f is a scalar function of (u1, u2, u3) while F = F1u1 +  F2u2 +  F3u3 is a vector function of

(u1, u2, u3).

            For the cylindrical polar coordinate system

            For the spherical polar coordinate system