Legendre function

For the most important case of integer degree, see Legendre polynomial and associated Legendre polynomial

In mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions P^μ
_λ, Q^μ
_λ are generalizations of Legendre polynomials to non-integer degree.

Differential equation

Associated Legendre functions are solutions of the Legendre equation

$(1-x^{2})\,y''-2xy'+\left(\lambda [\lambda +1]-{\frac {\mu ^{2}}{1-x^{2}}}\right)\,y=0,\,$

where the complex numbers λ and μ are called the degree and order of the associated Legendre functions respectively. Legendre functions are the associated Legendre functions of order μ=0.

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

Definition

These functions may actually be defined for general complex parameters and argument:

$P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}),\qquad {\text{for }}|1-z|<2$

where $\Gamma$ is the gamma function and $_{2}F_{1}$ is the hypergeometric function.

The second order differential equation has a second solution, $Q_{\lambda }^{\mu }(z)$, defined as:

$Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {1}{z^{\lambda +\mu +1}}}(1-z^{2})^{\mu /2}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right)$

Integral representaions

The Legendre functions can be written as contour integrals. For example

$P_{\lambda }(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt$

where the contour circles around the points 1 and z in the positive direction and does not circle around −1. For real x, x\geq1, we have <math>P_s(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta<math>

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 8". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 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. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publischer, Inc.
• Dunster, T. M. (2010), "Legendre and Related Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Ivanov, A.B. (2001) [1994], "Legendre function", Encyclopedia of Mathematics, EMS Press
• Snow, Chester (1952) [1942], Hypergeometric and Legendre functions with applications to integral equations of potential theory, National Bureau of Standards Applied Mathematics Series, No. 19, Washington, D.C.: U. S. Government Printing Office, MR0048145
• Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2 {{citation}}: ISBN / Date incompatibility (help)

External links

• Legendre function P on the Wolfram functions site.
• Legendre function Q on the Wolfram functions site.
• Associated Legendre function P on the Wolfram functions site.
• Associated Legendre function Q on the Wolfram functions site.