“Legendre Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Legendre Differential Equation

释义

Legendre Differential Equation

The second-order Ordinary Differential Equation

(1)

which can be rewritten

(2)

The above form is a special case of the associated Legendre differential equation with

. The Legendre differentialequation has Regular Singular Points at

, 1, and

. It can be solved using aseries expansion,

			(3)
			(4)
			(5)

Plugging in,

(6)


	(7)


	(8)


	(9)

(10)

so each term must vanish and

(11)


			(12)

Therefore,

			(13)

			(14)


			(15)

so the Even solution is

(16)

Similarly, the Odd solution is

(17)

If is an Even Integer, the series reduces to a Polynomial of degree with only EvenPowers of and the series diverges. If is an Odd Integer, the series reducesto a Polynomial of degree with only Odd Powers of and the series diverges. Thegeneral solution for an Integer is given by the Legendre Polynomials

(18)

where

is chosen so that

. If the variable

is replaced by

, then the Legendredifferential equation becomes

(19)

as is derived for the associated Legendre differential equation with

The associated Legendre differential equation is

(20)

(21)

The solutions to this equation are called the associated Legendre polynomials. Writing

, first establishthe identities

(22)

(23)


			(24)

and

(25)

Therefore,


			(26)

Plugging (22) into (26) and the result back into (21) gives

(27)

(28)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 332, 1972.

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