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

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

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

4. 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.

5. 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

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

7. 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:

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

9. Summary of Solution

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

11. Spherical Wave Functions (cont.)
We then have

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

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

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

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

16. Spherical Wave Functions (cont.)
Define
Then

17. 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!

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

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

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

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

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

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

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

25. Recurrence Relations
Denote:
We have the following relations:

26. Orthogonality Relation

27. Wronskians

28. Modified Spherical Bessel Functions
Different factors!

29. For More Information