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Associated Legendre Differential Equation

The associated Legendre differential equation is a generalization of the Legendre differential equation given by

d/(dx)[(1-x^2)(dy)/(dx)]+[l(l+1)-(m^2)/(1-x^2)]y=0,
(1)

which can be written

(1-x^2)(d^2y)/(dx^2)-2x(dy)/(dx)+[l(l+1)-(m^2)/(1-x^2)]y=0
(2)

(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions P_l^m(x) to this equation are called the associated Legendre polynomials (if l is an integer), or associated Legendre functions of the first kind (if l is not an integer). The complete solution is

y=C_1P_l^m(x)+C_2Q_l^m(x),
(3)

where Q_l^m(x) is a Legendre function of the second kind.

The associated Legendre differential equation is often written in a form obtained by setting x=costheta. Plugging the identities

(dy)/(dx)=(dy)/(d(costheta))
(4)
=-1/(sintheta)(dy)/(dtheta)
(5)
(d^2y)/(dx^2)=1/(sintheta)d/(dtheta)(1/(sintheta)(dy)/(dtheta))
(6)
=1/(sin^2theta)((d^2y)/(dtheta^2)-(costheta)/(sintheta)(dy)/(dtheta))
(7)

into (◇) then gives

((d^2y)/(dtheta^2)-(costheta)/(sintheta)(dy)/(dtheta))+2(costheta)/(sintheta)(dy)/(dtheta)+[l(l+1)-(m^2)/(sin^2theta)]y=0
(8)
(d^2y)/(dtheta^2)+(costheta)/(sintheta)(dy)/(dtheta)+[l(l+1)-(m^2)/(sin^2theta)]y=0.
(9)

See also

Associated Legendre Polynomial, Legendre Differential Equation, Legendre Function of the Second Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 332, 1972.Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.

Cite this as:

Weisstein, Eric W. "Associated Legendre Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AssociatedLegendreDifferentialEquation.html

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