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Legendre Differential Equation


The Legendre differential equation is the second-order ordinary differential equation

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

which can be rewritten

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

The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case m=0. The Legendre differential equation has regular singular points at -1, 1, and infty.

If the variable x is replaced by costheta, then the Legendre differential equation becomes

 (d^2y)/(dtheta^2)+(costheta)/(sintheta)(dy)/(dtheta)+l(l+1)y=0,
(3)

derived below for the associated (m!=0) case.

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution P_l(x) which is regular at finite points is called a Legendre function of the first kind, while a solution Q_l(x) which is singular at +/-1 is called a Legendre function of the second kind. If l is an integer, the function of the first kind reduces to a polynomial known as the Legendre polynomial.

The Legendre differential equation can be solved using the Frobenius method by making a series expansion with k=0,

y=sum_(n=0)^(infty)a_nx^n
(4)
y^'=sum_(n=0)^(infty)na_nx^(n-1)
(5)
y^('')=sum_(n=0)^(infty)n(n-1)a_nx^(n-2).
(6)

Plugging in,

 (1-x^2)sum_(n=0)^inftyn(n-1)a_nx^(n-2)-2xsum_(n=0)^inftyna_nx^(n-1)+l(l+1)sum_(n=0)^inftya_nx^n=0
(7)
sum_(n=0)^(infty)n(n-1)a_nx^(n-2)-sum_(n=0)^(infty)n(n-1)a_nx^n
(8)
 -2xsum_(n=0)^(infty)na_nx^(n-1)+l(l+1)sum_(n=0)^(infty)a_nx^n=0
(9)
sum_(n=2)^(infty)n(n-1)a_nx^(n-2)-sum_(n=0)^(infty)n(n-1)a_nx^n
(10)
 -2sum_(n=0)^(infty)na_nx^n+l(l+1)sum_(n=0)^(infty)a_nx^n=0
(11)
sum_(n=0)^(infty)(n+2)(n+1)a_(n+2)x^n-sum_(n=0)^(infty)n(n-1)a_nx^n
(12)
 -2sum_(n=0)^(infty)na_nx^n+l(l+1)sum_(n=0)^(infty)a_nx^n=0
(13)
 sum_(n=0)^infty{(n+1)(n+2)a_(n+2)+[-n(n-1)-2n+l(l+1)]a_n}=0,
(14)

so each term must vanish and

 (n+1)(n+2)a_(n+2)+[-n(n+1)+l(l+1)]a_n=0
(15)
a_(n+2)=(n(n+1)-l(l+1))/((n+1)(n+2))a_n
(16)
=-([l+(n+1)](l-n))/((n+1)(n+2))a_n.
(17)

Therefore,

a_2=-(l(l+1))/(1·2)a_0
(18)
a_4=-((l-2)(l+3))/(3·4)a_2
(19)
=(-1)^2([(l-2)l][(l+1)(l+3)])/(1·2·3·4)a_0
(20)
a_6=-((l-4)(l+5))/(5·6)a_4
(21)
=(-1)^3([(l-4)(l-2)l][(l+1)(l+3)(l+5)])/(1·2·3·4·5·6)a_0,
(22)

so the even solution is

 y_1(x)=1+sum_(n=1)^infty(-1)^n([(l-2n+2)...(l-2)l][(l+1)(l+3)...(l+2n-1)])/((2n)!)x^(2n).
(23)

Similarly, the odd solution is

 y_2(x)=x+sum_(n=1)^infty(-1)^n([(l-2n+1)...(l-3)(l-1)][(l+2)(l+4)...(l+2n)])/((2n+1)!)x^(2n+1).
(24)

If l is an even integer, the series y_1(x) reduces to a polynomial of degree l with only even powers of x and the series y_2(x) diverges. If l is an odd integer, the series y_2(x) reduces to a polynomial of degree l with only odd powers of x and the series y_1(x) diverges. The general solution for an integer l is then given by the Legendre polynomials

P_n(x)=c_n{y_1(x) for l even; y_2(x) for l odd
(25)
=c_n{_2F_1(-1/2,1/2(l+1);1/2,x^2) for l even; x_2F_1(1/2(l+2),1/2(1-l);3/2;x^2) for l odd
(26)

where c_n is chosen so as to yield the normalization P_n(1)=1 and _2F_1(a,b;c;z) is a hypergeometric function.

A generalization of the Legendre differential equation is known as the associated Legendre differential equation.

Moon and Spencer (1961, p. 155) call the differential equation

 (1-x^2)y^('')-2xy^'-[k^2a^2(x^2-1)-p(p+1)-(q^2)/(x^2-1)]y=0
(27)

the Legendre wave function equation (Zwillinger 1997, p. 124).


See also

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

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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.

Referenced on Wolfram|Alpha

Legendre Differential Equation

Cite this as:

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

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