Associated Legendre polynomials

When called associated Legendre polynomials and associated Legendre polynomials, and associated spherical functions, there are functions that are used in mathematics and theoretical physics. Since not all of the associated Legendre polynomials are truly, many authors speak of assigned or associated Legendrefunktionen.

The associated Legendre polynomials are the solutions of the general Legendregleichung:

This ordinary differential equation has non-singular solutions in the interval only if and are integers with.

Encountered general Legendregleichung (and thus the associated Legendre polynomials ) are often in physics, particularly when a spherical symmetry is present, such as the central potential. Here the Laplace equation and related partial differential equations can often be traced back to the general Legendregleichung. The most prominent example of this is the quantum- mechanical solution of the energy states of the hydrogen atom.

Definition

The associated Legendre polynomials are referred to as. The easiest way can be defined as derivatives of ordinary Legendre polynomials:

Where the -th Legendre polynomial is

It follows

Connection with Legendre polynomials

The generalized Legendregleichung goes for the Legendregleichung so true.

Orthogonality

For the associated Legendre polynomials are in the interval I = [ -1,1] two orthogonality relations:

The second integral is only defined if either m or n is not 0.

Related to the unit sphere

Most important is the case. The associated Legendre equation then

Since, according to the substitution rule

Applies to transfer above orthogonality readily on the unit sphere.

About the so-called spherical harmonics are defined as

Which form a complete orthonormal system on the unit sphere.

The first associated Legendre

For the associated Legendre following recurrence formula is valid

The first Legendre polynomials are determined to be so

And as an argument

Assigned Legendrefunktionen second kind

Similar to the Legendre equation, the associated Legendre polynomials only one set of solution functions of the generalized Legendre equation represents the associated Legendrefunktionen second kind as solutions provide dar. also applies to them with the Legendrefunktionen second kind.