Associated Legendre Differential Equation
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
30-5-2018
1118
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,](http://mathworld.wolfram.com/images/equations/AssociatedLegendreDifferentialEquation/NumberedEquation1.gif) |
(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](http://mathworld.wolfram.com/images/equations/AssociatedLegendreDifferentialEquation/NumberedEquation2.gif) |
(2)
|
(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions 
to this equation are called the associated Legendre polynomials (if 
is an integer), or associated Legendre functions of the first kind (if 
is not an integer). The complete solution is
 |
(3)
|
where 
is a Legendre function of the second kind.
The associated Legendre differential equation is often written in a form obtained by setting 
. Plugging the identities
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](http://mathworld.wolfram.com/images/equations/AssociatedLegendreDifferentialEquation/NumberedEquation4.gif) |
(8)
|
![(d^2y)/(dtheta^2)+(costheta)/(sintheta)(dy)/(dtheta)+[l(l+1)-(m^2)/(sin^2theta)]y=0.](http://mathworld.wolfram.com/images/equations/AssociatedLegendreDifferentialEquation/NumberedEquation5.gif) |
(9
|
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.
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