Legendre function

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title: "Legendre function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["hypergeometric-functions"] description: "Solutions of Legendre's differential equation" topic_path: "general/hypergeometric-functions" source: "https://en.wikipedia.org/wiki/Legendre_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Solutions of Legendre's differential equation ::

In physical science and mathematics, the Legendre functions P**λ, Q**λ and associated Legendre functions , , and Legendre functions of the second kind, Qn, 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.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/75/Associated_Legendre_Poly.svg" caption="1=''λ'' = ''l'' = 5}}."] ::

Legendre's differential equation

The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right] y = 0, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n), and are the Legendre polynomials Pn; and when λ is an integer (denoted n), and is also an integer with λ*, Qλ. However, the solution Qλ when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted Qn.

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 hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

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, 2F_1. With \Gamma being the gamma function, the first solution is 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| and the second is 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. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e0/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" caption="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" 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"] ::

These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if μ is non-zero. A useful relation between the P and Q solutions is Whipple's formula.

Positive integer order

For positive integer \mu = m \in \N^+ the evaluation of P^\mu_\lambda above involves cancellation of singular terms. We can find the limit valid for m \in \N_0 as

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),

with (\lambda)_{n} the (rising) Pochhammer symbol.

Legendre functions of the second kind ({{math|''Q_n''}})

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/84/Mplwp_legendreQ04.svg" caption="Plot of the first five Legendre functions of the second kind."] ::

The nonpolynomial solution for the special case of integer degree \lambda = n \in \N_0 , and \mu = 0 , is often discussed separately. It is given by 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)

This solution is necessarily singular when x = \pm 1 .

The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula 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}

Associated Legendre functions of the second kind

The nonpolynomial solution for the special case of integer degree \lambda = n \in \N_0 , and \mu = m \in \N_0 is given by Q_n^{m}(x) = (-1)^m (1-x^2)^\frac{m}{2} \frac{d^m}{dx^m}Q_n(x),.

Integral representations

The Legendre functions can be written as contour integrals. For example, 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 where the contour winds around the points 1 and z in the positive direction and does not wind around −1. For real x, we have 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

Legendre function as characters

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

Singularities of Legendre functions of the first kind ({{math|''P''_''λ''}}) as a consequence of symmetry

Legendre functions P**λ 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 Q**λ 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 that the singularity of the Legendre functions P**λ 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.

References

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References

1. (2018). "Fast generation of isotropic Gaussian random fields on the sphere". Monte Carlo Methods and Applications.
2. 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.

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