# Legendre polynomials

**Legendre polynomials** (aka. Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
are solutions of the Legendre differential equation.
The Legendre polynomial, can be defined by the contour integral

The first seven Legendre polynomials are:

"shifted" Legendre polynomials (which obey the orthogonality relationship):

Powers in terms of Legendre polynomials: