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Legendre function
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<h2 id="legendres-differential-equation">Legendre's differential equation</h2> <p>The general Legendre equation reads ( 1 − x 2 ) y ″ − 2 x y ′ + [ λ ( λ + 1 ) − μ 2 1 − x 2 ] y = 0 , {\displaystyle \left(1-x^{2}\right)y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]y=0,} where the numbers <i>λ</i> and <i>μ</i> may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when <i>λ</i> is an integer (denoted <i>n</i>), and <i>μ</i> = 0 are the Legendre polynomials <i>Pn</i>; and when <i>λ</i> is an integer (denoted <i>n</i>), and <i>μ</i> = <i>m</i> is also an integer with |<i>m</i>| < <i>n</i> are the associated Legendre polynomials. All other cases of <i>λ</i> and <i>μ</i> can be discussed as one, and the solutions are written <i>P</i><i>μ</i><i>λ</i>, <i>Q</i><i>μ</i><i>λ</i>. If <i>μ</i> = 0, the superscript is omitted, and one writes just <i>Pλ</i>, <i>Qλ</i>. However, the solution <i>Qλ</i> when <i>λ</i> is an integer is often discussed separately as Legendre's function of the second kind, and denoted <i>Qn</i>. </p><p>This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a <a href="/facts/Hypergeometric_differential_equation/yTEC6NBJ">hypergeometric differential equation</a> by a change of variable, and its solutions can be expressed using <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric functions</a>. </p> <h2 id="solutions-of-the-differential-equation">Solutions of the differential equation</h2> <p>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 <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a>, 2 F 1 {\displaystyle _{2}F_{1}} . With Γ {\displaystyle \Gamma } being the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>, the first solution is P λ μ ( z ) = 1 Γ ( 1 − μ ) [ z + 1 z − 1 ] μ / 2 2 F 1 ( − λ , λ + 1 ; 1 − μ ; 1 − z 2 ) , for | 1 − z | < 2 , {\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {z+1}{z-1}}\right]^{\mu /2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}\right),\qquad {\text{for }}\ |1-z|<2,} and the second is Q λ μ ( z ) = π Γ ( λ + μ + 1 ) 2 λ + 1 Γ ( λ + 3 / 2 ) e i μ π ( z 2 − 1 ) μ / 2 z λ + μ + 1 2 F 1 ( λ + μ + 1 2 , λ + μ + 2 2 ; λ + 3 2 ; 1 z 2 ) , for | z | > 1. {\displaystyle 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}}}\,_{2}F_{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.} </p> <p>These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if <i>μ</i> is non-zero. A useful relation between the <i>P</i> and <i>Q</i> solutions is <a href="/facts/Whipple_formulae/DVY8wfhw">Whipple's formula</a>. </p> <h3>Positive integer order</h3> <p>For positive integer μ = m ∈ N + {\displaystyle \mu =m\in \mathbb {N} ^{+}} the evaluation of P λ μ {\displaystyle P_{\lambda }^{\mu }} above involves cancellation of singular terms. We can find the limit valid for m ∈ N 0 {\displaystyle m\in \mathbb {N} _{0}} as<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p> P λ m ( z ) = lim μ → m P λ μ ( z ) = ( − λ ) m ( λ + 1 ) m m ! [ 1 − z 1 + z ] m / 2 2 F 1 ( − λ , λ + 1 ; 1 + m ; 1 − z 2 ) , {\displaystyle P_{\lambda }^{m}(z)=\lim _{\mu \to m}P_{\lambda }^{\mu }(z)={\frac {(-\lambda )_{m}(\lambda +1)_{m}}{m!}}\left[{\frac {1-z}{1+z}}\right]^{m/2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1+m;{\frac {1-z}{2}}\right),} </p><p>with ( λ ) n {\displaystyle (\lambda )_{n}} the (rising) <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a>. </p> <h2 id="legendre-functions-of-the-second-kind-qn">Legendre functions of the second kind (<i>Qn</i>)</h2> <p>The nonpolynomial solution for the special case of integer degree λ = n ∈ N 0 {\displaystyle \lambda =n\in \mathbb {N} _{0}} , and μ = 0 {\displaystyle \mu =0} , is often discussed separately. It is given by Q n ( x ) = n ! 1 ⋅ 3 ⋯ ( 2 n + 1 ) ( x − ( n + 1 ) + ( n + 1 ) ( n + 2 ) 2 ( 2 n + 3 ) x − ( n + 3 ) + ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) 2 ⋅ 4 ( 2 n + 3 ) ( 2 n + 5 ) x − ( n + 5 ) + ⋯ ) {\displaystyle Q_{n}(x)={\frac {n!}{1\cdot 3\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\cdot 4(2n+3)(2n+5)}}x^{-(n+5)}+\cdots \right)} </p><p>This solution is necessarily <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singular</a> when x = ± 1 {\displaystyle x=\pm 1} . </p><p>The Legendre functions of the second kind can also be defined recursively via <a href="/facts/Legendre_polynomials/nDvV4em0">Bonnet's recursion formula</a> Q n ( x ) = { 1 2 log 1 + x 1 − x n = 0 P 1 ( x ) Q 0 ( x ) − 1 n = 1 2 n − 1 n x Q n − 1 ( x ) − n − 1 n Q n − 2 ( x ) n ≥ 2 . {\displaystyle 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}}xQ_{n-1}(x)-{\frac {n-1}{n}}Q_{n-2}(x)&n\geq 2\,.\end{cases}}} </p> <h2 id="associated-legendre-functions-of-the-second-kind">Associated Legendre functions of the second kind</h2> <p>The nonpolynomial solution for the special case of integer degree λ = n ∈ N 0 {\displaystyle \lambda =n\in \mathbb {N} _{0}} , and μ = m ∈ N 0 {\displaystyle \mu =m\in \mathbb {N} _{0}} is given by Q n m ( x ) = ( − 1 ) m ( 1 − x 2 ) m 2 d m d x m Q n ( x ) . {\displaystyle Q_{n}^{m}(x)=(-1)^{m}(1-x^{2})^{\frac {m}{2}}{\frac {d^{m}}{dx^{m}}}Q_{n}(x)\,.} </p> <h2 id="integral-representations">Integral representations</h2> <p>The Legendre functions can be written as contour integrals. For example, P λ ( z ) = P λ 0 ( z ) = 1 2 π i ∫ 1 , z ( t 2 − 1 ) λ 2 λ ( t − z ) λ + 1 d t {\displaystyle P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt} where the contour winds around the points 1 and <i>z</i> in the positive direction and does not wind around −1. For real <i>x</i>, we have P s ( x ) = 1 2 π ∫ − π π ( x + x 2 − 1 cos θ ) s d θ = 1 π ∫ 0 1 ( x + x 2 − 1 ( 2 t − 1 ) ) s d t t ( 1 − t ) , s ∈ C {\displaystyle 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 \mathbb {C} } </p> <h2 id="legendre-function-as-characters">Legendre function as characters</h2> <p>The real integral representation of P s {\displaystyle P_{s}} are very useful in the study of harmonic analysis on L 1 ( G / / K ) {\displaystyle L^{1}(G//K)} where G / / K {\displaystyle G//K} is the <a href="/facts/Homogeneous_space/YGebqcJA">double coset space</a> of S L ( 2 , R ) {\displaystyle SL(2,\mathbb {R} )} (see <a href="/facts/Zonal_spherical_function/myzsFsrU">Zonal spherical function</a>). Actually the Fourier transform on L 1 ( G / / K ) {\displaystyle L^{1}(G//K)} is given by L 1 ( G / / K ) ∋ f ↦ f ^ {\displaystyle L^{1}(G//K)\ni f\mapsto {\hat {f}}} where f ^ ( s ) = ∫ 1 ∞ f ( x ) P s ( x ) d x , − 1 ≤ ℜ ( s ) ≤ 0 {\displaystyle {\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0} </p> <h2 id="singularities-of-legendre-functions-of-the-first-kind-p-as-a-consequence-of-symmetry">Singularities of Legendre functions of the first kind (<i>P</i><i>λ</i>) as a consequence of symmetry</h2> <p>Legendre functions <i>P</i><i>λ</i> 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 <i>Q</i><i>λ</i> of the second kind are always unbounded, in order to have a bounded solution of Legendre's equation at all, the degree <i>must</i> be integer valued: <i>only</i> for integer degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1] . It can be shown<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> that the singularity of the Legendre functions <i>P</i><i>λ</i> 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. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Ferrers_function/d3q79kkQ">Ferrers function</a></li></ul> <ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="http://www.math.ubc.ca/~cbm/aands/page_332.htm">"Chapter 8"</a>. <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 332. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li> <li><a href="/facts/Richard_Courant/QDToNfWl">Courant, Richard</a>; <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a> (1953), <i>Methods of Mathematical Physics, Volume 1</i>, New York: Interscience Publisher, Inc.</li> <li>Dunster, T. M. (2010), <a href="http://dlmf.nist.gov/14">"Legendre and Related Functions"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li> <li>Ivanov, A.B. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Legendre_function">"Legendre function"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li> <li>Snow, Chester (1952) [1942], <a href="http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346"><i>Hypergeometric and Legendre functions with applications to integral equations of potential theory</i></a>, National Bureau of Standards Applied Mathematics Series, No. 19, Washington, D.C.: U. S. Government Printing Office, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2027%2Fmdp.39015011416826">2027/mdp.39015011416826</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0048145">0048145</a></li> <li><a href="/facts/E._T._Whittaker/b7wl3DMJ">Whittaker, E. T.</a>; <a href="/facts/G._N._Watson/VLfFjgjh">Watson, G. N.</a> (1963), <i>A Course in Modern Analysis</i>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-58807-2</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://functions.wolfram.com/HypergeometricFunctions/LegendrePGeneral/">Legendre function P</a> on the Wolfram functions site.</li> <li><a href="http://functions.wolfram.com/HypergeometricFunctions/LegendreQGeneral/">Legendre function Q</a> on the Wolfram functions site.</li> <li><a href="http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/">Associated Legendre function P</a> on the Wolfram functions site.</li> <li><a href="http://functions.wolfram.com/HypergeometricFunctions/LegendreQ2General/">Associated Legendre function Q</a> on the Wolfram functions site.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Creasey, Peter E.; Lang, Annika (2018). "Fast generation of isotropic Gaussian random fields on the sphere". Monte Carlo Methods and Applications. 24 (1): 1–11. arXiv:1709.10314. Bibcode:2018MCMA...24....1C. doi:10.1515/mcma-2018-0001. S2CID 4657044. <a href="https://www.degruyter.com/document/doi/10.1515/mcma-2018-0001/html?lang=de" target="_blank">https://www.degruyter.com/document/doi/10.1515/mcma-2018-0001/html?lang=de</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>van der Toorn, Ramses (4 April 2022). "The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre's Equation". Symmetry. 14 (4): 741. Bibcode:2022Symm...14..741V. doi:10.3390/sym14040741. ISSN 2073-8994. <a href="https://doi.org/10.3390%2Fsym14040741" target="_blank">https://doi.org/10.3390%2Fsym14040741</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>