Editing Legendre function

{{Short description|Solutions of Legendre's differential equation}}
{{For|the most common case of integer degree|Legendre polynomials|associated Legendre polynomials}}
{{more footnotes|date=January 2013}}

In physical science and mathematics, the '''Legendre functions''' {{math|''P''<sub>''λ''</sub>}}, {{math|''Q''<sub>''λ''</sub>}} and '''associated Legendre functions''' {{math|''P''{{su|p=''μ''|b=''λ''}}}}, {{math|''Q''{{su|p=''μ''|b=''λ''}}}}, and '''Legendre functions of the second kind''', {{math|''Q<sub>n</sub>''}}, are all solutions of Legendre's differential equation. The [[Legendre polynomials]] and the [[associated Legendre polynomials]] are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles.

[[File:Associated Legendre Poly.svg|thumb|500px|Associated Legendre polynomial curves for {{math|1=''λ'' = ''l'' = 5}}.]]

== Legendre's differential equation ==
The '''general Legendre equation''' reads
<math display="block">\left(1 - x^2\right) y'' - 2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right] y = 0,</math>
where the numbers {{math|''λ''}} and {{math|''μ''}} may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when {{math|''λ''}} is an integer (denoted {{math|''n''}}), and {{math|1=''μ'' = 0}} are the Legendre polynomials {{math|''P<sub>n</sub>''}}; and when 
{{math|''λ''}} is an integer (denoted {{math|''n''}}), and {{math|1=''μ'' = ''m''}} is also an integer with {{math|{{abs|''m''}} < ''n''}} are the associated Legendre polynomials. All other cases of {{math|''λ''}} and {{math|''μ''}} can be discussed as one, and the solutions are written {{math|''P''{{su|p=''μ''|b=''λ''}}}}, {{math|''Q''{{su|p=''μ''|b=''λ''}}}}. If {{math|1=''μ'' = 0}}, the superscript is omitted, and one writes just {{math|''P<sub>λ</sub>''}}, {{math|''Q<sub>λ</sub>''}}. However, the solution {{math|''Q<sub>λ</sub>''}} when {{math|''λ''}} is an integer is often discussed separately as Legendre's function of the second kind, and denoted {{math|''Q<sub>n</sub>''}}.

This is a second order linear equation with three regular singular points (at {{math|1}}, {{math|−1}}, and {{math|∞}}). Like all such equations, it can be converted into a [[hypergeometric differential equation]] by a change of variable, and its solutions can be expressed using [[hypergeometric function]]s.

== Solutions of the differential equation ==
Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the [[hypergeometric function]], <math> _2F_1</math>. With <math>\Gamma</math> being the [[gamma function]], the first solution is
<math display="block">P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{z+1}{z-1}\right]^{\mu/2} \,_2F_1 \left(-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}\right),\qquad \text{for } \  |1-z|<2,</math>
and the second is
<math display="block">Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ |z|>1.</math>
[[File:Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if {{math|''μ''}} is non-zero. A useful relation between the {{math|''P''}} and {{math|''Q''}} solutions is [[Whipple formulae|Whipple's formula]].

===Positive integer order ===
For positive integer <math> \mu = m \in \N^+ </math> the evaluation of <math> P^\mu_\lambda </math> above involves cancellation of singular terms. We can find the limit valid for <math> m \in \N_0 </math> as<ref>{{cite journal | url=https://www.degruyter.com/document/doi/10.1515/mcma-2018-0001/html?lang=de | doi=10.1515/mcma-2018-0001 | title=Fast generation of isotropic Gaussian random fields on the sphere | date=2018 | last1=Creasey | first1=Peter E. | last2=Lang | first2=Annika | journal=Monte Carlo Methods and Applications | volume=24 | issue=1 | pages=1–11 | arxiv=1709.10314 | bibcode=2018MCMA...24....1C | s2cid=4657044 }}</ref>

<math display="block">P^m_\lambda(z) = \lim_{\mu \to m} P^\mu_\lambda (z) = \frac{(-\lambda )_m (\lambda + 1)_m}{m!} \left[\frac{1-z}{1+z}\right]^{m/2} \,_2F_1 \left(-\lambda, \lambda+1; 1+m; \frac{1-z}{2}\right), </math>

with <math>(\lambda)_{n}</math> the (rising) [[Pochhammer symbol]].

==Legendre functions of the second kind ({{math|''Q<sub>n</sub>''}})==

[[File:Mplwp_legendreQ04.svg|thumb|384px|Plot of the first five Legendre functions of the second kind.]]

The nonpolynomial solution for the special case of integer degree  <math> \lambda = n \in \N_0 </math>, and <math> \mu = 0 </math>, is often discussed separately. 
It is given by
<math display="block">Q_n(x)=\frac{n!}{1\cdot3\cdots(2n+1)}\left(x^{-(n+1)}+\frac{(n+1)(n+2)}{2(2n+3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2\cdot4(2n+3)(2n+5)}x^{-(n+5)}+\cdots\right)</math>

This solution is necessarily [[Singularity (mathematics)|singular]] when <math> x = \pm 1 </math>.

The Legendre functions of the second kind can also be defined recursively via [[Legendre polynomials#Definition via generating function|Bonnet's recursion formula]]
<math display="block">Q_n(x) = \begin{cases}
 \frac{1}{2} \log \frac{1+x}{1-x}  & n = 0  \\
 P_1(x) Q_0(x) - 1  & n = 1  \\
 \frac{2n-1}{n} x Q_{n-1}(x) - \frac{n-1}{n} Q_{n-2}(x)  & n \geq 2 \,.
\end{cases}</math>

== Associated Legendre functions of the second kind ==
The nonpolynomial solution for the special case of integer degree <math> \lambda = n \in \N_0 </math>, and <math> \mu = m \in \N_0 </math> is given by
<math display="block">Q_n^{m}(x) = (-1)^m (1-x^2)^\frac{m}{2} \frac{d^m}{dx^m}Q_n(x)\,.</math>

==Integral representations==
The Legendre functions can be written as contour integrals. For example,
<math display="block">P_\lambda(z) =P^0_\lambda(z) = \frac{1}{2\pi i}
 \int_{1,z} \frac{(t^2-1)^\lambda}{2^\lambda(t-z)^{\lambda+1}}dt</math>
where the contour winds around the points {{math|1}} and {{math|''z''}} in the positive direction and does not wind around {{math|−1}}.
For real {{math|''x''}}, we have
<math display="block">P_s(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta = \frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\right)^s\frac{dt}{\sqrt{t(1-t)}},\qquad s\in\Complex</math>

==Legendre function as characters==
The real integral representation of <math>P_s</math> are very useful in the study of harmonic analysis on <math>L^1(G//K)</math> where <math>G//K</math> is the [[Homogeneous space|double coset space]] of <math>SL(2,\R)</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by
<math display="block">L^1(G//K)\ni f\mapsto \hat{f}</math>
where
<math display="block">\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 </math>

==Singularities of Legendre functions of the first kind ({{math|''P''<sub>''λ''</sub>}}) as a consequence of symmetry ==
Legendre functions {{math|''P''<sub>''λ''</sub>}} of non-integer degree are unbounded at the interval [-1, 1] . In applications in physics, this often provides a selection criterion. Indeed, because Legendre functions {{math|''Q''<sub>''λ''</sub>}} of the second kind are always unbounded, in order to have a bounded solution of Legendre's equation at all, the degree ''must'' be integer valued: ''only'' for integer degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1] . It can be shown<ref>{{Cite journal |last=van der Toorn |first=Ramses |date=4 April 2022 |title=The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre's Equation |journal=Symmetry |language=en |volume=14 |issue=4 |pages=741 |doi=10.3390/sym14040741 |bibcode=2022Symm...14..741V |issn=2073-8994 |doi-access=free }}</ref> that the singularity of the Legendre functions  {{math|''P''<sub>''λ''</sub>}}  for non-integer degree is a consequence of the mirror symmetry of Legendre's equation. Thus there is a symmetry under the selection rule just mentioned.

== See also ==
* [[Ferrers function]]

==References==
{{Reflist}}

* {{AS ref|8|332}}
* {{citation|first1=Richard|last1=Courant|author-link1=Richard Courant|first2=David|last2=Hilbert|author-link2=David Hilbert |year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publisher, Inc|location=New York}}.
*{{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
*{{eom|id=L/l058030|first=A.B.|last= Ivanov}}
*{{Citation | last1=Snow | first1=Chester | title=Hypergeometric and Legendre functions with applications to integral equations of potential theory | url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 | publisher=U. S. Government Printing Office | location=Washington, D.C. | series=National Bureau of Standards Applied Mathematics Series, No. 19 | mr=0048145 | year=1952|orig-year=1942| hdl=2027/mdp.39015011416826 | hdl-access=free }}
*{{Citation | last1=Whittaker | first1=E. T. |author-link1=E. T. Whittaker| last2=Watson | first2=G. N. | author-link2=G. N. Watson |title=A Course in Modern Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-58807-2 | year=1963 }}

==External links==
*[http://functions.wolfram.com/HypergeometricFunctions/LegendrePGeneral/ Legendre function P] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQGeneral/ Legendre function Q] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/ Associated Legendre function P] on the Wolfram functions site.
*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQ2General/ Associated Legendre function Q] on the Wolfram functions site.

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[[Category:Hypergeometric functions]]