Associated Legendre function: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Paul Wormer
imported>Paul Wormer
(32 intermediate revisions by 4 users not shown)
Line 1: Line 1:
In [[mathematics]] and [[physics]], an '''associated Legendre function'''  ''P''<sub>''l''</sub><sup>(''m'')</sup> is related to a [[Legendre polynomial]] ''P''<sub>''l''</sub> by the following equation
{{subpages}}
{{TOC|right}}
 
:*''See [[Associated Legendre function/Catalogs]] for explicit equations through''  ''ℓ'' = 6.
 
In [[mathematics]] and [[physics]], an '''associated Legendre function'''  ''P''<sub>''''</sub><sup>''m''</sup> is related to a [[Legendre polynomial]] ''P''<sub>''''</sub> by the following equation
:<math>
:<math>
P^{(m)}_\ell(x) = (1-x^2)^{m/2} \frac{d P_\ell(x)}{dx^\ell}.
P^{m}_\ell(x) = (1-x^2)^{m/2} \frac{d^m P_\ell(x)}{dx^m}, \qquad 0 \le m \le \ell.
</math>
</math>
For even ''m'' the associated Legendre function is a polynomial, for odd ''m'' the function contains the factor (1-''x'' &sup2; )<sup>&frac12;</sup> and hence is not a polynomial.  
Although extensions are possible, in this article ''ℓ'' and ''m'' are restricted to integer numbers.  For even ''m'' the associated Legendre function is a polynomial, for odd ''m'' the function contains the factor (1&minus;''x'' &sup2; )<sup>&frac12;</sup> and hence is not a polynomial.
 
The associated Legendre functions are important in [[quantum mechanics]] and [[potential theory]].


The associated Legendre polynomials are important in [[quantum mechanics]] and [[potential theory]].
According to Ferrers<ref> N. M. Ferrers, ''An Elementary Treatise on Spherical Harmonics'', MacMillan, 1877 (London),  p. 77. [http://www.archive.org/stream/elementarytreati00ferriala#page/2/mode/2up Online].</ref> the polynomials were named  "Associated Legendre functions" by the British mathematician [[Isaac Todhunter]] in 1875,<ref>I. Todhunter, ''An Elementary Treatise on Laplace's, Lamé's, and Bessel's Functions'', MacMillan, 1875 (London).  In fact, Todhunter called the Legendre polynomials "Legendre coefficients". </ref> where "associated function" is Todhunter's translation of the German term ''zugeordnete Function'', coined in 1861 by  [[Heinrich Eduard Heine|Heine]],<ref> E. Heine, ''Handbuch der Kugelfunctionen'', G. Reimer, 1861 (Berlin).[http://books.google.com/books?id=YE8DAAAAQAAJ&pg=PA3&dq=Eduard+Heine&hl=en#PPR1,M1 Google book online]</ref> and "Legendre"  is in honor of the French mathematician [[Adrien-Marie Legendre]] (1752–1833), who was the first to introduce and study the functions.


==Differential equation==
==Differential equation==
Define
Define
:<math>
:<math>
\Pi^{(m)}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m},
\Pi^{m}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m},
</math>
</math>
where ''P''<sub>''l''</sub>(''x'')  is a Legendre polynomial.
where ''P''<sub>''''</sub>(''x'')  is a Legendre polynomial.
Differentiating the [[Legendre polynomials#differential equation|Legendre differential equation]]:  
Differentiating the [[Legendre polynomials#differential equation|Legendre differential equation]]:
:<math>
:<math>
(1-x^2) \frac{d^2 \Pi^{(0)}_\ell(x)}{dx^2} - 2 x \frac{d\Pi^{(0)}_\ell(x)}{dx} + \ell(\ell+1)  
(1-x^2) \frac{d^2 \Pi^{0}_\ell(x)}{dx^2} - 2 x \frac{d\Pi^{0}_\ell(x)}{dx} + \ell(\ell+1)
\Pi^{(0)}_\ell(x) = 0,
\Pi^{0}_\ell(x) = 0,
</math>
</math>
''m'' times gives an equation for &Pi;<sup>(''m'')</sup><sub>''l''</sub>
''m'' times gives an equation for &Pi;<sup>''m''</sup><sub>''l''</sub>
:<math>
:<math>
(1-x^2) \frac{d^2 \Pi^{(m)}_\ell(x)}{dx^2} - 2(m+1) x \frac{d\Pi^{(m)}_\ell(x)}{dx} + \left[\ell(\ell+1)  
(1-x^2) \frac{d^2 \Pi^{m}_\ell(x)}{dx^2} - 2(m+1) x \frac{d\Pi^{m}_\ell(x)}{dx} + \left[\ell(\ell+1)
-m(m+1) \right] \Pi^{(m)}_\ell(x) = 0  .
-m(m+1) \right] \Pi^{m}_\ell(x) = 0  .
</math>
</math>
After substitution of  
After substitution of
:<math>
:<math>
\Pi^{(m)}_\ell(x) = (1-x^2)^{-m/2} P^{(m)}_\ell(x),
\Pi^{m}_\ell(x) = (1-x^2)^{-m/2} P^{m}_\ell(x),
</math>
</math>
we find, after multiplying through with <math>(1-x^2)^{m/2}</math>, that the ''associated Legendre differential equation'' holds for the associated Legendre functions
and after multiplying through with <math>(1-x^2)^{m/2}</math>, we find the ''associated Legendre differential equation'':
:<math>
:<math>
(1-x^2) \frac{d^2 P^{(m)}_\ell(x)}{dx^2} -2x\frac{d P^{(m)}_\ell(x)}{dx} +
(1-x^2) \frac{d^2 P^{m}_\ell(x)}{dx^2} -2x\frac{d P^{m}_\ell(x)}{dx} +
\left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{(m)}_\ell(x).
\left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{m}_\ell(x)= 0 .
</math>
</math>
In physical applications usually ''x'' = cos&theta;, then then associated Legendre differential equation takes the form
One often finds the equation written in the following equivalent way
:<math>
:<math>
\frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{(m)}_\ell
\left( (1-x^{2})\; y\,' \right)' +\left( \ell(\ell+1)
+\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{(m)}_\ell.
-\frac{m^{2} }{1-x^{2} } \right) y=0, 
</math>
</math>
where the primes indicate differentiation with respect to ''x''.
In physical applications it is usually the case that  ''x'' = cos&theta;, then the  associated Legendre differential equation takes the form
:<math>
\frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{m}_\ell
+\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{m}_\ell = 0.
</math>
==Extension to negative m==
==Extension to negative m==
By the [[Legendre polynomials#Rodrigues formula|Rodrigues]]  formula, one obtains
By the [[Legendre polynomials#Rodrigues formula|Rodrigues]]  formula, one obtains


:<math>P_\ell^{(m)}(x) = \frac{1}{2^\ell \ell!} (1-x^2)^{m/2}\  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell.</math>
:<math>P_\ell^{m}(x) = \frac{1}{2^\ell \ell!} (1-x^2)^{m/2}\  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell.</math>


This equation allows extension of the range of ''m'' to: -''l'' &le; ''m'' &le; ''l''.  
This equation allows extension of the range of ''m'' to: &minus;''m'' &le; '''' &le; ''m''.


Since the associated Legendre equation is invariant under the substitution ''m'' &rarr; -''m'', the equations for ''P''<sub>''l''</sub><sup>( &plusmn;''m'')</sup>, resulting from this expression, are proportional.  
Since the associated Legendre equation is invariant under the substitution ''m'' &rarr; &minus;''m'', the equations for ''P''<sub>''''</sub><sup> &plusmn;''m''</sup>, resulting from this expression, are proportional.<ref>The associated Legendre differential equation being of second order, the general solution is of the form  <math>AP_\ell^m + BQ_\ell^m</math> where <math>Q_\ell^m</math> is a Legendre polynomial of the second kind, which has a singularity at ''x'' = 0. Hence solutions that are regular at ''x'' = 0 have ''B'' = 0 and are proportional to <math>P_\ell^m</math>.  The Rodrigues formula shows that <math>P_\ell^{-m}</math> is a regular (at ''x''=0)  solution and the proportionality follows.</ref>


To obtain the proportionality constant we consider
To obtain the proportionality constant we consider
:<math>
:<math>
(1-x^2)^{-m/2} \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^{m/2}  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad  0 \le m \le \ell,  
(1-x^2)^{-m/2} \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^{m/2}  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad  0 \le m \le \ell,
</math>
</math>
and we bring the factor (1-''x''&sup2;)<sup>-''m''/2</sup> to the other side.  
and we bring the factor (1&minus;''x''&sup2;)<sup>&minus;''m''/2</sup> to the other side.
Equate the coefficients of the same powers of ''x'' on the left and right hand side of  
Equate the coefficient of the highest power of ''x'' on the left and right hand side of
:<math>
:<math>
\frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad  0 \le m \le \ell,  
\frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad  0 \le m \le \ell,
</math>
</math>
and it follows that the proportionality constant is
and it follows that the proportionality constant is
:<math>
:<math>
c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,
c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad  0 \le m \le \ell,
</math>
so that the associated Legendre functions of same |''m''| are related to each other by
:<math>
P^{-|m|}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{|m|}_\ell(x).
</math>
Note that the phase factor (&minus;1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1&minus;''x''&sup2;)<sup>m</sup>.
 
==Orthogonality relations==
Important integral relations are:
:<math>\int\limits_{-1}^{1}P_{l}^{m} \left( x\right) P_{k}^{m} \left( x\right)
dx =\frac{2}{2l+1} \frac{\left( l+m\right) !}{\left( l-m\right) !} \delta
_{lk}, </math>
and:
:<math>
\int_{-1}^{1} P^{m}_{\ell}(x) P^{n}_{\ell}(x) \frac{d x}{1-x^2} =
\frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!},  \qquad  m \ne 0  .
</math>
The latter integral for ''n'' = ''m'' = 0
:<math>
\int_{-1}^{1} P^{0}_{\ell}(x) P^{0}_{\ell}(x) \frac{d x}{1-x^2}
</math>
</math>
so that the associated Legendre functions of same |''m''| are related to each other by
is undetermined (infinite). (see the subpage [[Associated_Legendre_function/Proofs|Proofs]] for detailed proofs of these relations.)
 
==Recurrence relations==
 
The functions satisfy the following difference equations, which are taken from Edmonds.<ref>A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', Princeton University Press, 2nd edition (1960)</ref>
 
:<math>
(\ell-m+1)P_{\ell+1}^{m}(x) - (2\ell+1)xP_{\ell}^{m}(x) + (\ell+m)P_{\ell-1}^{m}(x)=0
</math> <!-- Edmonds 2.5.20 -->
 
:<math>
xP_{\ell}^{m}(x) -(\ell-m+1)(1-x^2)^{1/2} P_{\ell}^{m-1}(x) - P_{\ell-1}^{m}(x)=0
</math> <!-- Edmonds 2.5.21 -->
 
:<math>
P_{\ell+1}^{m}(x) - x P_{\ell}^{m}(x)-(\ell+m)(1-x^2)^{1/2}P_{\ell}^{m-1}(x)=0
</math> <!-- Edmonds 2.5.22 -->
 
:<math>
(\ell-m+1)P_{\ell+1}^{m}(x)+(1-x^2)^{1/2}P_{\ell}^{m+1}(x)-
(\ell+m+1) xP_{\ell}^{m}(x)=0
</math><!-- Edmonds 2.5.23 -->
 
:<math>
:<math>
P^{(-|m|)}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{(|m|)}_\ell(x).
(1-x^2)^{1/2}P_{\ell}^{m+1}(x)-2mxP_{\ell}^{m}(x)+
(\ell+m)(\ell-m+1)(1-x^2)^{1/2}P_{\ell}^{m-1}(x)=0
</math><!-- Edmonds 2.5.24 -->
 
:<math>
(1-x^2)\frac{dP_{\ell}^{m}}{dx}(x) =(\ell+1)xP_{\ell}^{m}(x) -(\ell-m+1)P_{\ell+1}^{m}(x)
</math>
</math>
Note that the phase factor (-1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1-''x''&sup2;)<sup>m</sup>.
:::::::<math>
=(\ell+m)P_{\ell-1}^{m}(x)-\ell x P_{\ell}^{m}(x)
</math><!-- Edmonds 2.5.25 -->
 
==Reference==
<references />

Revision as of 07:58, 24 January 2010

This article is developed but not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
Catalogs [?]
Proofs [?]
 
This editable, developed Main Article is subject to a disclaimer.

In mathematics and physics, an associated Legendre function Pm is related to a Legendre polynomial P by the following equation

Although extensions are possible, in this article and m are restricted to integer numbers. For even m the associated Legendre function is a polynomial, for odd m the function contains the factor (1−x ² )½ and hence is not a polynomial.

The associated Legendre functions are important in quantum mechanics and potential theory.

According to Ferrers[1] the polynomials were named "Associated Legendre functions" by the British mathematician Isaac Todhunter in 1875,[2] where "associated function" is Todhunter's translation of the German term zugeordnete Function, coined in 1861 by Heine,[3] and "Legendre" is in honor of the French mathematician Adrien-Marie Legendre (1752–1833), who was the first to introduce and study the functions.

Differential equation

Define

where P(x) is a Legendre polynomial. Differentiating the Legendre differential equation:

m times gives an equation for Πml

After substitution of

and after multiplying through with , we find the associated Legendre differential equation:

One often finds the equation written in the following equivalent way

where the primes indicate differentiation with respect to x.

In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form

Extension to negative m

By the Rodrigues formula, one obtains

This equation allows extension of the range of m to: −mm.

Since the associated Legendre equation is invariant under the substitution m → −m, the equations for P ±m, resulting from this expression, are proportional.[4]

To obtain the proportionality constant we consider

and we bring the factor (1−x²)m/2 to the other side. Equate the coefficient of the highest power of x on the left and right hand side of

and it follows that the proportionality constant is

so that the associated Legendre functions of same |m| are related to each other by

Note that the phase factor (−1)m arising in this expression is not due to some arbitrary phase convention, but arises from expansion of (1−x²)m.

Orthogonality relations

Important integral relations are:

and:

The latter integral for n = m = 0

is undetermined (infinite). (see the subpage Proofs for detailed proofs of these relations.)

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds.[5]

Reference

  1. N. M. Ferrers, An Elementary Treatise on Spherical Harmonics, MacMillan, 1877 (London), p. 77. Online.
  2. I. Todhunter, An Elementary Treatise on Laplace's, Lamé's, and Bessel's Functions, MacMillan, 1875 (London). In fact, Todhunter called the Legendre polynomials "Legendre coefficients".
  3. E. Heine, Handbuch der Kugelfunctionen, G. Reimer, 1861 (Berlin).Google book online
  4. The associated Legendre differential equation being of second order, the general solution is of the form where is a Legendre polynomial of the second kind, which has a singularity at x = 0. Hence solutions that are regular at x = 0 have B = 0 and are proportional to . The Rodrigues formula shows that is a regular (at x=0) solution and the proportionality follows.
  5. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)