1. Notes are from D. R. Wilton, Dept. of ECE
Fall 2016
David R. Jackson
Notes 20
Bessel Functions
Notes are from D. R. Wilton, Dept. of ECE

2. Cylindrical Wave Functions
Helmholtz equation:
In cylindrical coordinates:
Separation of variables:
let
Substitute into previous equation and divide by .

3. Cylindrical Wave Functions (cont.)
Divide by 
let

4. Cylindrical Wave Functions (cont.)
(1)
Therefore
Hence, f (z) = constant = - kz2

5. Cylindrical Wave Functions (cont.)
Hence
Next, to isolate the  -dependent term, multiply Eq. (1) by  2 :

6. Cylindrical Wave Functions (cont.)
Hence
(2)
Hence so

7. Cylindrical Wave Functions (cont.)
From Eq. (2) we then have
The next goal is to solve this equation for R().
First, multiply by R and collect terms:

8. Cylindrical Wave Functions (cont.)
Define
Then,
Next, define
Note that
and

9. Cylindrical Wave Functions (cont.)
Then we have
Bessel equation of order 
Two independent solutions:
Hence
Therefore

10. Cylindrical Wave Functions (cont.)
Summary

11. References for Bessel Functions
M. R. Spiegel, Schaum’s Outline Mathematical Handbook, McGraw-Hill, 4th Edition, 2012.
M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Government Printing Office, Tenth Printing, 1972.
NIST online Digital Library of Mathematical Functions (
N. N. Lebedev, Special Functions & Their Applications, Dover Publications, New York, 1972.
G. N. Watson, “A Treatise on the Theory of Bessel Functions” (2nd Ed.), Cambridge University Press, 1995.

12. Properties of Bessel Functions
Jn (x) x

13. Bessel Functions (cont.)
Yn (x) x

14. Bessel Functions (cont.)
Frobenius solution:
This series always converges.

15. Bessel Functions (cont.)
Non-integer order:
Two linearly independent solutions
Bessel equation is unchanged by
Note:
is a always a valid solution
These are linearly independent when  is not an integer.

16. Bessel Functions (cont.)
Bessel function of second kind:
  …-2, -1, 0, 1, 2 …
(This definition gives a “nice” asymptotic behavior as x  .)

17. Bessel Functions (cont.)
From the Frobenius solution we have:
(Schaum’s Outline Mathematical Handbook, Eq. (24.9))
where

18. Bessel Functions (cont.)
Integer order:
Symmetry property:
(They are no longer linearly independent.)

19. Bessel Functions (cont.)
Small-Argument Properties (x  0):
Examples:
For order zero, the Bessel function of the second kind behaves logarithmically rather than algebraically.

20. Bessel Functions (cont.)
Asymptotic Formulas:

21. Hankel Functions Incoming wave Outgoing wave
These are valid for arbitrary order .

22. Hankel Functions (cont.)
Useful identity:
This is a symmetry property of the Hankel function.
N. N. Lebedev, Special Functions & Their Applications, Dover Publications, New York, 1972.

23. Generating Function
The integer order Bessel function of the first kind can also be defined through a generating function g(x,t):
The generating function definition leads to a number of useful identities and representations:

24. Recurrence Relations
Many recurrence relations can be derived from the generating function.
Equating like powers of t yields:

25. Recurrence Relations (cont.)
This can be used to generate other useful recurrence relations:

26. Recurrence Relations (cont.)
Also, we have
Equating like powers of t yields:

27. Recurrence Relations (cont.)
Then use
This can be used to replace Jn+1 or Jn-1.
This yields

28. Recurrence Relations (cont.)
The same recurrence formulas actually apply to all Bessel functions of all orders.
If Z(x) denotes any Bessel, Neumann, or Hankel function of order , then we have:
Integral relations also follow from these (see the next slide).

29. Recurrence Relations (cont.)
Example of integral identity:
Hence
Similarly, we have

30. Recurrence Relations (cont.)
Example
(First one,  = 1)
(Second one,  = 0)

31. Wronskians
From the Sturm-Liouville properties, the Wronskians for the Bessel differential equation are found to have the following form:
The constant C can be found using the small-argument approximations for the Bessel functions.
Note: For   n, the Wronskian is not identically zero (in fact, it is not zero anywhere), and hence the two functions are linearly independent.

32. Wronskians
Similarly, we have

33. Fourier-Bessel Series
Note:
The order  is arbitrary here.
To find the coefficients, use the orthogonality identity (derivation on next slide):
Note:
See Notes 18 for a derivation of orthogonality when m  n.

34. Fourier-Bessel Series (cont.)

35. Fourier-Bessel Series (cont.)
Derivation of the orthogonality formula
Start with this integral identity:
Hence we have

36. Addition Theorems
Addition theorems allow cylindrical harmonics in one coordinate system to be expanded in terms of those of a shifted coordinate system.
Shifting from global origin to local origin:

37. Addition Theorems (cont.)
Shifting from local origin to global origin:

38. Addition Theorems (cont.)
Special case (n = 0):