Spherical Bessel Functions ECE 6382 Fall 2016 David R. Jackson Notes 23 Spherical Bessel Functions

Spherical Wave Functions Consider solving in spherical coordinates. y x z

Spherical Wave Functions (cont.) In spherical coordinates we have Hence we have Using separation of variables, let

Spherical Wave Functions (cont.) After substituting Eq. (2) into Eq. (1), divide by : At this point, we cannot yet say that all of the dependence on a given variable is within only one term.

Spherical Wave Functions (cont.) Next, multiply by r2 sin2  : Since the underlined term is the only one which depends on , It must be equal to a constant, Hence, set

Spherical Wave Functions (cont.) Now divide Eq. (3) by and use Eq. (4), to obtain

Spherical Wave Functions (cont.) The underlined terms are the only ones that involve  now. This time the separation constant is customarily chosen as –n(n+1). We then have:

Spherical Wave Functions (cont.) Substituting Eq. (6) into Eq. (5), the differential equation for the radial function R is then

Summary of Solution

Spherical Bessel Functions Consider the differential equation for the radial function R: Make the following substitution: Also, denote

Spherical Wave Functions (cont.) We then have

Spherical Wave Functions (cont.) (self-adjoint form) or “spherical Bessel equation” Solution: bn (x) Note the lower case b.

Spherical Wave Functions (cont.) Denote and let Hence

Spherical Wave Functions (cont.) Multiply by Combine these terms Combine these terms or Use Define

Spherical Wave Functions (cont.) We then have This is Bessel’s equation of order . Hence so that added for convenience

Spherical Wave Functions (cont.) Define Then

Properties of Spherical Bessel Functions Integer order (n = integer): Bessel functions of half-integer order are given by closed-form expressions. This becomes a closed-form expression!

Properties of Spherical Bessel Functions (cont.) Examples:

Properties of Spherical Bessel Functions (cont.) Proof for  = 1/2 Start with: Hence

Properties of Spherical Bessel Functions (cont.) Examine the factorial expression: Note:

Properties of Spherical Bessel Functions (cont.) Therefore Hence, we have

Properties of Spherical Bessel Functions (cont.) or We then recognize that

Properties of Spherical Bessel Functions (cont.) In general, we have the following closed-form representations: where

Properties of Spherical Bessel Functions (cont.) Other closed-form representations:

Recurrence Relations Denote: We have the following relations:

Orthogonality Relation

Wronskians

Modified Spherical Bessel Functions Different factors!

For More Information http://functions.wolfram.com/Bessel-TypeFunctions/