Associated Legendre function: Difference between revisions
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In [[mathematics]] and [[physics]], an '''associated Legendre function''' ''P''<sub>'' | :*''See [[Associated Legendre function/Catalogs]] for explicit equations through'' ''ℓ'' = 6. | ||
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 | |||
:<math> | :<math> | ||
P^{ | 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 | 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−''x'' ² )<sup>½</sup> and hence is not a polynomial. | ||
The associated Legendre functions are important in [[quantum mechanics]] and [[potential theory]]. | The associated Legendre functions 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^{ | \Pi^{m}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m}, | ||
</math> | </math> | ||
where ''P''<sub>'' | where ''P''<sub>''ℓ''</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^{ | (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^{ | \Pi^{0}_\ell(x) = 0, | ||
</math> | </math> | ||
''m'' times gives an equation for Π<sup> | ''m'' times gives an equation for Π<sup>''m''</sup><sub>''l''</sub> | ||
:<math> | :<math> | ||
(1-x^2) \frac{d^2 \Pi^{ | (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(m+1) \right] \Pi^{m}_\ell(x) = 0 . | ||
</math> | </math> | ||
After substitution of | After substitution of | ||
:<math> | :<math> | ||
\Pi^{ | \Pi^{m}_\ell(x) = (1-x^2)^{-m/2} P^{m}_\ell(x), | ||
</math> | </math> | ||
and after multiplying through with <math>(1-x^2)^{m/2}</math>, we find the ''associated Legendre differential equation'': | 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^{ | (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^{ | \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{m}_\ell(x)= 0 . | ||
</math> | |||
One often finds the equation written in the following equivalent way | |||
:<math> | |||
\left( (1-x^{2})\; y\,' \right)' +\left( \ell(\ell+1) | |||
-\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'' = cosθ, then the associated Legendre differential equation takes the form | In physical applications it is usually the case that ''x'' = cosθ, then the associated Legendre differential equation takes the form | ||
:<math> | :<math> | ||
\frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{ | \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^{ | +\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{m}_\ell = 0. | ||
</math> | </math> | ||
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By the [[Legendre polynomials#Rodrigues formula|Rodrigues]] formula, one obtains | By the [[Legendre polynomials#Rodrigues formula|Rodrigues]] formula, one obtains | ||
:<math>P_\ell^{ | :<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: | This equation allows extension of the range of ''m'' to: −''m'' ≤ ''ℓ'' ≤ ''m''. | ||
Since the associated Legendre equation is invariant under the substitution ''m'' → −''m'', the equations for ''P''<sub>'' | Since the associated Legendre equation is invariant under the substitution ''m'' → −''m'', the equations for ''P''<sub>''ℓ''</sub><sup> ±''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''²)<sup>−''m''/2</sup> to the other side. | and we bring the factor (1−''x''²)<sup>−''m''/2</sup> to the other side. | ||
Equate the coefficient of the highest power 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 | ||
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c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad 0 \le m \le \ell, | c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad 0 \le m \le \ell, | ||
</math> | </math> | ||
so that the associated Legendre functions of same |''m''| are related to each other by | so that the associated Legendre functions of same |''m''| are related to each other by | ||
:<math> | :<math> | ||
P^{ | P^{-|m|}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{|m|}_\ell(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''²)<sup>m</sup>. | 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''²)<sup>m</sup>. | ||
==Orthogonality relations== | ==Orthogonality relations== | ||
Important integral relations are | 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> | :<math> | ||
\int_{-1}^{1} P^{ | \int_{-1}^{1} P^{m}_{\ell}(x) P^{n}_{\ell}(x) \frac{d x}{1-x^2} = | ||
\frac{ | \frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!}, \qquad m \ne 0 . | ||
</math> | </math> | ||
The latter integral for ''n'' = ''m'' = 0 | |||
:<math> | :<math> | ||
\int_{-1}^{1} P^{ | \int_{-1}^{1} P^{0}_{\ell}(x) P^{0}_{\ell}(x) \frac{d x}{1-x^2} | ||
</math> | </math> | ||
is undetermined (infinite). (see the subpage [[Associated_Legendre_function/Proofs|Proofs]] for detailed proofs of these relations.) | |||
==Recurrence 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> | 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> | :<math> | ||
(\ell-m+1)P_{\ell+1}^{ | (\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> <!-- Edmonds 2.5.20 --> | ||
:<math> | :<math> | ||
xP_{\ell}^{ | 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> <!-- Edmonds 2.5.21 --> | ||
:<math> | :<math> | ||
P_{\ell+1}^{ | 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> <!-- Edmonds 2.5.22 --> | ||
:<math> | :<math> | ||
(\ell-m+1)P_{\ell+1}^{ | (\ell-m+1)P_{\ell+1}^{m}(x)+(1-x^2)^{1/2}P_{\ell}^{m+1}(x)- | ||
(\ell+m+1) xP_{\ell}^{ | (\ell+m+1) xP_{\ell}^{m}(x)=0 | ||
</math><!-- Edmonds 2.5.23 --> | </math><!-- Edmonds 2.5.23 --> | ||
:<math> | :<math> | ||
(1-x^2)^{1/2}P_{\ell}^{ | (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}^{ | (\ell+m)(\ell-m+1)(1-x^2)^{1/2}P_{\ell}^{m-1}(x)=0 | ||
</math><!-- Edmonds 2.5.24 --> | </math><!-- Edmonds 2.5.24 --> | ||
:<math> | :<math> | ||
(1-x^2)\frac{dP_{\ell}^{ | (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> | ||
:::::::<math> | :::::::<math> | ||
=(\ell+m)P_{\ell-1}^{ | =(\ell+m)P_{\ell-1}^{m}(x)-\ell x P_{\ell}^{m}(x) | ||
</math><!-- Edmonds 2.5.25 --> | </math><!-- Edmonds 2.5.25 --> | ||
==Reference== | ==Reference== | ||
<references /> | <references /> | ||
Revision as of 07:58, 24 January 2010
- See Associated Legendre function/Catalogs for explicit equations through ℓ = 6.
In mathematics and physics, an associated Legendre function Pℓm is related to a Legendre polynomial Pℓ by the following equation
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−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 Pℓ(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: −m ≤ ℓ ≤ m.
Since the associated Legendre equation is invariant under the substitution m → −m, the equations for Pℓ ±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
- ↑ N. M. Ferrers, An Elementary Treatise on Spherical Harmonics, MacMillan, 1877 (London), p. 77. Online.
- ↑ 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".
- ↑ E. Heine, Handbuch der Kugelfunctionen, G. Reimer, 1861 (Berlin).Google book online
- ↑ 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.
- ↑ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)