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

单词

Spherical Bessel Differential Equation

释义

Spherical Bessel Differential Equation

Take the Helmholtz Differential Equation

(1)

in Spherical Coordinates. This is just Laplace's Equation in Spherical Coordinates with an additionalterm,

(2)

Multiply through by

(3)

This equation is separable in

. Call the separation constant

(4)

Now multiply through by

(5)

This is the Spherical Bessel Differential Equation. It can be transformed by letting

, then

(6)

Similarly,

(7)

so the equation becomes

(8)

Now look for a solution of the form

, denoting a derivative with respect to

by a prime,

			(9)

			(10)



	(11)

(12)

(13)

(14)

(15)

But the solutions to this equation are Bessel Functions of half integral order, so the normalizedsolutions to the original equation are

(16)

which are known as Spherical Bessel Functions. The two types of solutions are denoted

(Spherical Bessel Function of the First Kind) or

(Spherical Bessel Function of the SecondKind), and the general solution is written

(17)

where

			(18)
			(19)

See also Spherical Bessel Function, Spherical Bessel Function of the First Kind, Spherical Bessel Function ofthe Second Kind

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 437, 1972.

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