Special functions MOC

Legendre polynomial

The th Legendre polynomial for is a polynomial of degree given by the Rodrigues’ formula¹ 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

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Footnotes

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