## How are the Legendre polynomials and associated Legendre functions related?

Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions . {\\displaystyle [-1,1]} . That is, ∫ − 1 1 P m ( x ) P n ( x ) d x = 0 if n ≠ m . {\\displaystyle \\int _ {-1}^ {1}P_ {m} (x)P_ {n} (x)\\,dx=0\\quad { ext {if }}n eq m.}

### When did Adrien Marie Legendre start to use polynomials?

Adrien-Marie Legendre (September 18, 1752 – January 10, 1833) began using, what are now referred to as Legendre polynomials in 1784 while studying the attraction of spheroids and ellipsoids. His work was important for geodesy. 1. Legendre’s Equation and Legendre Functions The second order diﬀerential equation given as (1− x2) d2y dx2 −2x dy dx

**Where do Legendre polynomials occur in Laplace’s equation?**

Legendre polynomials occur in the solution of Laplace’s equation of the static potential, ∇2 Φ (x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle).

**How are Legendre polynomials expanded with spherical harmonics?**

Additional properties of Legendre polynomials. The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using where the unit vectors r and r′ have spherical coordinates (θ,φ) and (θ′,φ′), respectively.

## How to calculate Legendre polynomials in self adjoint form?

This follows from the general Sturm-Liouville problem. Put Legendre’s equation in self adjoint form; d dx [(1− x2) dPl(x) dx ] +l(l +1)Pl(x) = 0 Then look at the equation for Pn(x) and subtract the equations for Pland Pnafter multipli- cation of the ﬁrst by Pnand the later by Pl. Integrate the result between ±1. This results in [(1− x2)P nP