1. 14. Bessel Functions Bessel Functions of the 1st Kind, J (x)
Orthogonality
Neumann Functions, Bessel Functions of the 2nd Kind
Hankel Functions, I (x) and K (x)
Asymptotic Expansions
Spherical Bessel Functions

2. Defining Properties of Special Functions
Differential eq.
Series form / Generating function.
Recurrence relations.
Integral representation.
Ref :
M.Abramowitz & I.A.Stegun, Handbook of Mathematical Functions, Dover Publ. (1970)
NIST Digital Library of Mathematical Functions:
Basic Properties :
Orthonormality.
Asymptotic form.

3. Usage of Bessel Functions
Solutions to equations involving the Laplacian, 2 , in
circular cylindrical coordinates : Bessel / Modified Bessel functions
or spherical coordinates : Spherical Bessel functions

4. 1. Bessel Functions of the 1st Kind, J (x)
Bessel functions are Frobenius solutions of the Bessel ODE
for   1, 2, 3, …
(eq.7.48)
cf. gen. func.
1st kind Jn (x) : n = 0, 1, 2, 3, …
regular at x = 0.
Periodic with amp  x 1/2 as x  .
Mathematica

5. Generating Function for Integral Order
For n  0
( in eq.7.48 )
n = m < 0 : 
Generalize:

6. Recurrence    

7. Ex

8. Bessel’s Differential Equation
Any set of functions Z (x) satisfying the recursions must also satisfy the ODE,
though not necessarily the series expansion.
Proof : 

9.  
QED 

10. Integral Representation : Integral Order
C encloses t = 0.
n = integers
C = unit circle centered at origin : 
Re :
Im : 
n = integers

11. n = integers

12. Zeros of Bessel Functions
nk : kth zero of Jn(x)
Mathematica
nk : kth zero of Jn(x)
kth zero of J0(x) = kth zero of J1(x)
kth zero of Jn(x) ~ kth zero of Jn-1(x)

13. Example 14.1.1. Fraunhofer Diffraction, Circular Aperture
Kirchhoff's diffraction formula (scalar amplitude of field) :
Fraunhofer diffraction (far field) for incident plane wave, circular aperture : 
Mathematica

14. Primes on variables dropped for clarity. 
Intensity:
1st min: 
Mathematica

15. Example 14.1.2. Cylindrical Resonant Cavity
Wave equation in vacuum : 
Circular cylindrical cavity, axis along z-axis :
// means tangent to wall S
TM mode :  

16.    
mj = jth zero of Jm(x) .
resonant frequency
with 

17. Bessel Functions of Nonintegral Order
Formally, gives only Jn of integral order.
with
However, the series expansion can be extended to J of nonintegral order :
for   1, 2, 3, …
Caution: are linearly independent.

18. Schlaefli Integral C encloses t = 0. n = integers
For nonintegral  , is multi-valued.
Possible candidate for is
Strategy for proving
Show F satisfies Bessel eq. for J .
Show for x  0.
Mathematica

19. Consider any open contour C that doesn’t cross the branch cut

20.  this F is a solution of the Bessel eq.
For C1 :
 this F is a solution of the Bessel eq.
For
C = spatial inversion of C ,
( same as that for  ; B.cut. on +axis ) .
Set :
Mathematica
QED

21. 2. Orthogonality 
where
i.e., Z (k) is the eigenfunction, with eigenvalue k2 , of the operator L .
( Sturm-Liouville eigenvalue probem )
Helmholtz eq. in cylindrical coordinates
 with  
mn = nth root of Jm

22. is not self-adjoint 
is self-adjoint
J is an eigenfunction of L with eigenvalue k2. 
 L is Hermitian, i.e., , if the inner product is defined as

23.  

24. Orthogonal Sets Let  Orthogonality : orthogonal set Let 

25. Mathematica

26. Normalization 

27.  
Mathematica
Similarly :
(see Ex )

28. Bessel Series : J (  i  / a )
For any well-behaved function f () with f (a)  0 :
for any  > 1
with

29. Bessel Series J (  i  / a )
For any well-behaved function f () with f (a)  0 :
for any  > 1
with

30. Example 14.2.1. Electrostatic Potential: Hollow Cylinder
Hollow cylinder : axis // z-axis, ends at z = 0, h ; radius = a.
Potentials at boundaries :
Electrostatics, no charges :
Eigenstates with cylindrical symmetry :    

31.    

32. 3. Neumann Functions, Bessel Functions of the 2nd Kind
x << 1  

33.   Ex
agrees with
x << 1 

34. /2 phase difference with Jn
Mathematica
For x  ,
periodic with amp  x 1/2
/2 phase difference with Jn

35. Integral RepresentationEx Ex

36. Recurrence Relations  

37. 
Since Y satisfy the same RRs for J , they are also the solutions to the Bessel eq. 
Caution: Since RR relates solutions to different ODEs (of different ), it depends on their normalizations.

38. Wronskian Formulas For an ODE in self-adjoint form Ex.7.6.1
the Wronskian of any two solutions satisfies
Bessel eq. in self-adjoint form :
 For a noninteger  , the two independent solutions J & J satisfy

39. A can be determined at any point, such as x = 0.  

40. More Recurrence Relations
Combining the Wronskian with the previous recurrence relations, one gets many more recurrence relations

41. Uses of Neumann Functions
Complete the general solutions.
Applicable to any region excluding the origin ( e.g., coaxial cable, quantum scattering ).
Build up the Hankel functions ( for propagating waves ).

42. Example 14.3.1. Coaxial Wave Guides
EM waves in region between 2 concentric cylindrical conductors of radii a & b. ( c.f., Eg & Ex )
For TM mode in cylindrical cavity (eg ) :
For TM mode in coaxial cable of radii a & b :
with
Note: No cut-off for TEM modes.