14. Bessel Functions 1.Bessel Functions of the 1 st Kind, J (x) 2.Orthogonality 3.Neumann Functions, Bessel Functions of the 2 nd Kind 4.Hankel Functions, I (x) and K (x) 5.Asymptotic Expansions 6.Spherical Bessel Functions

Defining Properties of Special Functions 1. Differential eq. 2. Series form / Generating function. 3. Recurrence relations. 4. Integral representation. Basic Properties : 1. Orthonormality. 2. Asymptotic form. Ref : 1.M.Abramowitz & I.A.Stegun, Handbook of Mathematical Functions, Dover Publ. (1970) nd_stegun.pdf. nd_stegun.pdf 2.NIST Digital Library of Mathematical Functions:

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

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

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

Recurrence    

Ex

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 : 

  QED 

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

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

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

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

with      mj = j th zero of J m (x).  resonant frequency

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

Strategy for proving 1. Show F satisfies Bessel eq. for J. 2. Show for x  0. For nonintegral, is multi-valued. Possible candidate for is Schlaefli Integral C encloses t = 0. n = integers Mathematica

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

Set: For  C  = spatial inversion of C , ( same as that for  ; B.cut. on +axis ). QED Mathematica For C 1 :  this F is a solution of the Bessel eq.

2.Orthogonality  where i.e., Z (k  ) is the eigenfunction, with eigenvalue k 2, of the operator L. ( Sturm-Liouville eigenvalue probem ) Helmholtz eq.in cylindrical coordinates  with    mn = n th root of J m

 L is Hermitian, i.e.,, if the inner product is defined as is not self-adjoint  is self-adjoint  J is an eigenfunction of L with eigenvalue k 2.

 

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

Mathematica

Normalization 

  Mathematica Similarly : (see Ex )

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

Bessel SeriesJ (  i  / a ) For any well-behaved function f (  ) with f (a)  0 : for any >  1 with

Example 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 :    

   