Special functions MOC

Legendre polynomial

The th Legendre polynomial for is a polynomial of degree given by the Rodrigues' formula1 #m/def/fun

and is even or odd depending on the parity of .

Mathematica

The Legendre polynomial be generated in Wolfram Mathematica with LegendreP[ℓ, x].

Properties

  1. The Legendre polynomials satisfy the orthonormality condition
Proof of 1

Without loss of generality, assume

Now the integral term on the final line is zero, since the highest power of is and . Each of the sum terms contains at least one factor and is hence zero. Thus for the integral is zero. For the case of

Let , so , , and . Then

which proves ^P1

#state/develop | #lang/en | #SemBr

Footnotes

  1. 2018. Introduction to quantum mechanics, §4.1, p. 135 ↩