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

Recurrence Relations 

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

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

Proof of   

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

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

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

Mathematica

Recurrence Relations for I (x)  

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

Integral Representations Ex

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

§10.1   Ex 

5.Asymptotic Expansions 1.Expansion in negative powers [ Stokes’ method (Ex ) ]. Problem : Relation to named functions not known. 2. Steepest descent.

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



Expansion of an Integral Representation for K Proof : 1. R satisfies the MBE. Consider

 QED

Proof : 2. R = K for z  0. Let   QED

Proofs 1 & 2  R = K i.e. Proof : 3. K (z) decays exponentially for large z.  QED

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



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 :

  

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

Mathematica

Example 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

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

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

Pochhammer symbol where 



j n & y n Mathematica

   



For any Bessel functions F (x) = J (x), Y (x), H (1,2) (x) : Recurrence Relations For any spherical Bessel functions f n (x) = j n (x), y n (x), h n (1,2) (x) :    

 

Rayleigh Formulas Proof is by induction.

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

Limiting Values : x << 1 For x << 1 :

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

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

Zeros of Spherical Bessel Functions  nk : k th zero of j n (x)  nk : k th zero of j n (x) Mathematica k th zero of j 0 (x) = k th zero of J 1 (x) k th zero of j n (x) ~ k th zero of j n-1 (x)

Example 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

Ex More Orthogonality : General remarks : 1. Spatial confinement  energy quantization. 2. Finite zero-point energy ( uncertainty principle ). 3. E is angular momentum dependent. 4. Eigenfunction belonging to same l but different n are orthogonal.

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

Recurrence Relations 

i 0 (x), i 1 (x), i 2 (x), k 0 (x), k 1 (x), k 2 (x)

Mathematica

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

Example 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 : V 0 < E < 0  Numerical solution  Smooth connection :