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

2.   Ex
agrees with
x << 1 

3. /2 phase difference with Jn
Mathematica
For x  ,
periodic with amp  x 1/2
/2 phase difference with Jn

4. Integral RepresentationEx Ex

5. Recurrence Relations  

6. 
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.

7. 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

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

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

10. 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 ).

11. 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.

12. 4. Hankel Functions, H(1) (x) & H(2) (x)
Hankel functions of the 1st & 2nd kind :
c.f.
for x real
For x << 1,  > 0 : 

13. Recurrence Relations 

14. Contour Representations
See Schlaefli integral 
The integral representation
is a solution of the Bessel eq. if at end points of C. 

15. The integral representation
is a solution of the Bessel eq. for any C with end points t = 0 and Re t = .
Consider
Mathematica 
If one can prove
then

16. Proof of   

17. QED
i.e.
are saddle points.
(To be used in asymptotic expansions.)

18. 5. Modified Bessel Functions, I (x) & K (x)
Bessel equation : 
oscillatory
Modified Bessel equation : 
Modified Bessel functions
exponential
 Bessel eq.  Modified Bessel eq.
 are all solutions of the MBE.

19. I (x) Modified Bessel functions of the 1st kind :
I (x) is regular at x = 0 with  

20. Mathematica

21. Recurrence Relations for I (x) 

22. 2nd Solution K (x)
Modified Bessel functions of the 2nd kind ( Whitaker functions ) :
Recurrence relations :
For x  0 : Ex

23. Integral RepresentationsEx

24. Example 14.5.1. A Green’s Function
Green function for the Laplace eq. in cylindrical coordinates :
Let

25. 
§   Ex