ECE 6382 Fall 2019 David R Jackson Notes

ECE 6382 Fall 2019 David R. Jackson Notes 23 Spherical Bessel Functions 1

Spherical Wave Functions Consider solving in spherical coordinates. z y x 2

Spherical Wave Functions (cont. ) In spherical coordinates we have Hence we have Using separation of variables, let 3

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. 4

Spherical Wave Functions (cont. ) Next, multiply by r 2 sin 2 : Since the underlined term is the only one which depends on It must be equal to a constant, , Hence, set 5

Spherical Wave Functions (cont. ) Now divide Eq. (3) by and use Eq. (4), to obtain 6

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: 7

Spherical Wave Functions (cont. ) Substituting Eq. (6) into Eq. (5), the differential equation for the radial function R is then 8

Summary of Solution 9

Spherical Bessel Functions Consider the differential equation for the radial function R: Make the following substitution: Also, denote 10

Spherical Wave Functions (cont. ) We then have 11

Spherical Wave Functions (cont. ) (self-adjoint form) or “spherical Bessel equation” Solution: bn (x) Note the lower case b. 12

Spherical Wave Functions (cont. ) Denote and let Hence 13

Spherical Wave Functions (cont. ) Multiply by Combine these terms or Use Define 14

Spherical Wave Functions (cont. ) We then have This is Bessel’s equation of order . Hence so that added for convenience 15

Spherical Wave Functions (cont. ) Define Then 16

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! 17

Properties of Spherical Bessel Functions (cont. ) Examples: 18

Properties of Spherical Bessel Functions (cont. ) Proof for = 1/2 Start with: Hence 19

Properties of Spherical Bessel Functions (cont. ) Examine the factorial expression: Note: 20

Properties of Spherical Bessel Functions (cont. ) Therefore Hence, we have 21

Properties of Spherical Bessel Functions (cont. ) or We then recognize that 22

Properties of Spherical Bessel Functions (cont. ) In general, we have the following closed-form representations: where 23

Properties of Spherical Bessel Functions (cont. ) Other closed-form representations: 24

Recurrence Relations Denote: We have the following relations: 25

Orthogonality Relation 26

Wronskians 27

Modified Spherical Bessel Functions Different factors! 28

For More Information http: //functions. wolfram. com/Bessel-Type. Functions/ 29