1. 5. Asymptotic Expansions
Expansion in negative powers [ Stokes’ method (Ex ) ].
Problem : Relation to named functions not known.
Steepest descent.

2. Asymptotic Forms of H Contour integral representation:
Method of steepest descent ( §12.7 ) : 

3. 

4. Expansion of an Integral Representation for K
Consider
Proof : R satisfies the MBE.

5. 
QED

6. Proof : R = K for z  0.
Let  
QED

7. Proofs 1 & 2  R = K i.e.
Proof : K (z) decays exponentially for large z. 
QED

8. is a divergent asymptotic series
Series terminates for  
 z =  is an essential singularity
 No convergent series solution about z = .

9. 

10. Additional Asymptotic Forms
Asymptotic forms of other Bessel functions can be expressed in terms of P & Q . 
Analytic continued to all z
Analytic continued to all z :

11.   

12. Properties of the Asymptotic Forms
All Bessel functions have the asymptotic form
where
e.g.
good for

13. Mathematica

14. Example 14.6.1. Cylindrical Traveling Waves
Eg : 2-D vibrating circular membrane  standing waves
Consider 2-D vibrating circular membrane without boundary
 travelinging waves
For large r
Circular symmetry (no  dependence ) : 
diverges at r = 0

15. 6. Spherical Bessel Functions
Radial part of the Helmholtz eq. in spherical coordinates  
Spherical Bessel functions 

16. Definitions
Spherical Bessel functions ( integer orders only ) :  

17. where
Pochhammer symbol 

18. 

19. jn & yn
Mathematica

20.    

21. 

23. Recurrence Relations For any Bessel functions
F (x) = J (x) , Y (x) , H (1,2)(x) :
For any spherical Bessel functions
fn (x) = jn (x) , yn(x) , hn(1,2)(x) :    

24.  

25. Rayleigh Formulas
Proof is by induction.

26. Proof of Rayleigh Formula
For n = 1 :
Assuming case n to be true,
QED

27. Limiting Values : x << 1
For x << 1 :

28. Limiting Values : x >> n ( n + 1 ) / 2
Standing spherical waves 
Travelling spherical waves

29. Orthogonality & Zeros Set   r .
Note: n i for jn is numerically the same as n+1/2, i for Jn+1/2, . 

30. Zeros of Spherical 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)

31. Example 14.7.1. Particle in a Sphere
Schrodinger eq. for free particle of mass m in a sphere of radius a :
with 
Radial eq. for r  a : 
R is regular at r = 0  B = 0
quantized     

32. General remarks :
Spatial confinement  energy quantization.
Finite zero-point energy ( uncertainty principle ).
E is angular momentum dependent.
Eigenfunction belonging to same l but different n are orthogonal.
More Orthogonality : Ex

33. Modified Spherical Bessel Functions
Spherical Bessel equation :
Modified Spherical Bessel equation : 
Caution : 

34. Recurrence Relations 

35. i0(x), i1(x), i2(x), k0(x), k1(x), k2(x)

36. Mathematica

37. Limiting Values
For x << 1 :
For x >> 1 :

38. Example 14.7.2. Particle in a Finite Spherical Well
Schrodinger eq. for free particle of mass m in a well of radius a :
with 
Radial eq. :
Bound states :
V0 < E < 0
Smooth connection : 
Numerical solution 