Associated Legendre polynomials

When associated Legendre polynomials and associated Legendre , also associated spherical functions called, is functions , which in mathematics and theoretical physics are used. Since not all assigned Legendre polynomials are really polynomials , many authors also speak of assigned or associated Legendre functions .

The assigned Legendre polynomials are the solutions to the general Legendre equation:

${\displaystyle (1-x^{2})\,{\frac {\mathrm {d} ^{2}\,y}{\mathrm {d} x^{2}}}-2x{\frac {\mathrm {d} y}{\mathrm {d} x}}+\left(\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right)\,y=0}$

This ordinary differential equation has non-singular solutions in the interval only if and are integral with . ${\displaystyle [-1,1]}$ ${\displaystyle \ell \,}$ ${\displaystyle m\,}$ ${\displaystyle 0\leq m\leq \ell }$

The general Legendre equation (and thus the associated Legendre polynomials) is often encountered in physics, especially when there is spherical symmetry , such as in the central potential . Here the Laplace equation and related partial differential equations can often be reduced to the general Legendre equation . The most prominent example of this is the quantum mechanical solution of the energy states of the hydrogen atom .

definition

The assigned Legendre polynomials for m = 0 are the usual Legendre polynomials.

Assigned Legendre polynomials for m = 1

Assigned Legendre polynomials for m = 2

Assigned Legendre polynomials for m = 3

The associated Legendre polynomials are referred to as . The easiest way to define them as derivatives of ordinary Legendre polynomials : ${\displaystyle P_{\ell }^{(m)}(x)}$

${\displaystyle P_{\ell }^{(m)}(x)=(-1)^{m}\left(1-x^{2}\right)^{m/2}{\frac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}P_{\ell }(x)}$

where the -th is Legendre polynomial ${\displaystyle P_{\ell }(x)}$ ${\displaystyle \ell }$

${\displaystyle P_{\ell }(x)={\frac {1}{2^{\ell }\,\ell !}}\,{\frac {\mathrm {d} ^{\ell }}{\mathrm {d} x^{\ell }}}\left(x^{2}-1\right)^{\ell }}$.

This results in

${\displaystyle P_{\ell }^{(m)}(x)={\frac {(-1)^{m}}{2^{\ell }\,\ell !}}\left(1-x^{2}\right)^{m/2}{\frac {\mathrm {d} ^{\ell +m}}{\mathrm {d} x^{\ell +m}}}\left(x^{2}-1\right)^{\ell }.}$

Connection with Legendre polynomials

The generalized Legendre equation goes over into the Legendre equation so that . ${\displaystyle m=0}$ ${\displaystyle P_{\ell }^{(0)}(x)=P_{\ell }(x)}$

Orthogonality

For the assigned Legendre polynomials, two orthogonality relations apply in the interval : ${\displaystyle I=[-1,1]}$

${\displaystyle \int \limits _{-1}^{+1}P_{\ell }^{(m)}(x)\,P_{k}^{(m)}(x)\,\mathrm {d} x={\frac {2}{2\,\ell +1}}\,{\frac {(\ell +m)!}{(\ell -m)!}}\,\delta _{\ell k}.}$

${\displaystyle \int \limits _{-1}^{+1}P_{\ell }^{(m)}(x)\,P_{\ell }^{(n)}(x)\cdot {\frac {1}{1-x^{2}}}\,\mathrm {d} x={\frac {(\ell +m)!}{m(\ell -m)!}}\,\delta _{mn}.}$

The second integral is only defined if either or not equal to 0. ${\displaystyle m}$ ${\displaystyle n}$

Connection with the unit sphere

Most important is the case . The associated Legendre equation then reads ${\displaystyle x=\cos \vartheta }$

${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} \vartheta ^{2}}}+{\frac {\cos \vartheta }{\sin \vartheta }}\,{\frac {\mathrm {d} y}{\mathrm {d} \vartheta }}+\left[\ell \,(\ell +1)-{\frac {m^{2}}{\sin ^{2}\vartheta }}\right]y=0.}$

Because according to the substitution rule

${\displaystyle \int _{0}^{\pi }f(\cos \vartheta )\sin \vartheta \,\mathrm {d} \vartheta =\int _{-1}^{1}f(x)\mathrm {d} x}$

holds, the above orthogonality relations are easily transferred to the unit sphere.

The so-called spherical surface functions are defined as ${\displaystyle P_{\ell }^{(m)}(\cos \vartheta )}$

${\displaystyle Y_{\ell }^{(m)}(\varphi ,\vartheta )={\sqrt {{\frac {2\,\ell +1}{4\,\pi }}\,{\frac {(\ell -m)!}{(\ell +m)!}}}}\,P_{\ell }^{(m)}(\cos \vartheta )\,\mathrm {e} ^{i\,m\,\varphi },}$

which form a complete orthonormal system on the unit sphere.

The first assigned Legendre polynomials

The following recursion formula applies to the assigned Legendre polynomials

${\displaystyle (\ell -m)\,P_{\ell }^{(m)}(x)=x\,(2\,\ell -1)\,P_{\ell -1}^{(m)}(x)-(\ell +m-1)\,P_{\ell -2}^{(m)}(x).}$

The corresponding start values ​​of the recursion formula are shown as follows:

${\displaystyle P_{m}^{(m)}(x)=(-1)^{m}\cdot {\frac {(2m)!}{2^{m}m!}}\cdot \left(1-x^{2}\right)^{m/2}\quad ,\quad P_{k}^{m}(x)=0\;,\quad \forall k<m}$

The first Legendre polynomials are thus determined

${\displaystyle P_{0}^{(0)}(x)=1\!\,}$

${\displaystyle P_{1}^{(0)}(x)=x\!\,}$

${\displaystyle P_{1}^{(1)}(x)=-{\sqrt {1-x^{2}}}}$

${\displaystyle P_{2}^{(0)}(x)={\frac {1}{2}}\,(3\,x^{2}-1)}$

${\displaystyle P_{2}^{(1)}(x)=-3\,x{\sqrt {1-x^{2}}}}$

${\displaystyle P_{2}^{(2)}(x)=3\,(1-x^{2})}$

And with as an argument ${\displaystyle \cos \vartheta }$

${\displaystyle P_{0}^{(0)}(\cos \vartheta )=1}$

${\displaystyle P_{1}^{(0)}(\cos \vartheta )=\cos \vartheta }$

${\displaystyle P_{1}^{(1)}(\cos \vartheta )=-\sin \vartheta }$

${\displaystyle P_{2}^{(0)}(\cos \vartheta )={\frac {1}{2}}\,(3\,\cos ^{2}\vartheta -1)}$

${\displaystyle P_{2}^{(1)}(\cos \vartheta )=-3\,\sin \vartheta \,\cos \vartheta }$

${\displaystyle P_{2}^{(2)}(\cos \vartheta )=3\,\sin ^{2}\vartheta }$

Assigned Legendre functions of the 2nd kind

Similar to the Legendre equation, the assigned Legendre polynomials represent only a group of solution functions of the generalized Legendre equation. The assigned Legendre functions of the 2nd type also represent solutions. The same applies to them with the Legendre functions of the 2nd type . ${\displaystyle P_{\ell }^{(m)}(x)}$ ${\displaystyle Q_{\ell }^{(m)}(x)}$ ${\displaystyle Q_{\ell }^{(0)}=Q_{\ell }}$ ${\displaystyle Q_{\ell }(x)}$

Web links

• Legendre functions in the NIST Digital Library of Mathematical Functions (English)
• Eric W. Weisstein : Associated Legendre Polynomial . In: MathWorld (English).

literature

• Richard Courant , David Hilbert : Methods of Mathematical Physics . 2 volumes. Springer Verlag, 1968
• Gerald Teschl : Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators . American Mathematical Society, 2009 ( mat.univie.ac.at )