5. Asymptotic Expansions Expansion in negative powers [ Stokes’ method (Ex 14.6.10.) ]. Problem : Relation to named functions not known. Steepest descent.

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



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

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



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 14.6.1. Cylindrical Traveling Waves Eg. 14.1.24 : 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 ) :  

where Pochhammer symbol 



jn & yn Mathematica

   



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) :    

 

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  Standing spherical waves  Travelling spherical waves

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

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)

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     

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.14.7.12-3

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

Recurrence Relations 

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

Mathematica

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

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 