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14. Bessel Functions. Bessel Functions of the 1 st Kind, J ( x ) Orthogonality Neumann Functions, Bessel Functions of the 2 nd Kind Hankel Functions, I ( x ) and K ( x ) Asymptotic Expansions Spherical Bessel Functions. Defining Properties of Special Functions.
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14. Bessel Functions Bessel Functions of the 1st Kind, J(x) Orthogonality Neumann Functions, Bessel Functions of the 2nd Kind Hankel Functions, I(x) and K(x) Asymptotic Expansions Spherical Bessel Functions
Defining Properties of Special Functions • Differential eq. • Series form / Generating function. • Recurrence relations. • Integral representation. • Ref : • M.Abramowitz & I.A.Stegun, Handbook of Mathematical Functions, Dover Publ. (1970) http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf. • NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/ • Basic Properties : • Orthonormality. • Asymptotic form.
Usage of Bessel Functions Solutions to equations involving the Laplacian, 2 , in circular cylindrical coordinates : Bessel / Modified Bessel functions or spherical coordinates : Spherical Bessel functions
1. Bessel Functions of the 1st Kind, J(x) Bessel functions are Frobenius solutions of the Bessel ODE for 1, 2, 3, … (eq.7.48) cf. gen. func. 1st kindJn(x) : n= 0, 1, 2, 3, … regular at x = 0. Periodic with amp x 1/2 as x . Mathematica
Generating Function for Integral Order Generating function : For n 0 ( in eq.7.48 ) n= m < 0 : Generalize:
Recurrence
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:kthzero of Jn(x) Mathematica nk:kthzero of Jn(x) kthzero of J0(x) = kthzero of J1(x) kthzero of Jn(x) ~ kthzero of Jn-1(x)
Example 14.1.1. Fraunhofer Diffraction, Circular Aperture Kirchhoff's diffraction formula (scalar amplitude of field) : Fraunhofer diffraction (far field) for incident plane wave, circular aperture: Mathematica
Primes on variables dropped for clarity. Intensity: 1st min: Mathematica
Example 14.1.2. Cylindrical Resonant Cavity Wave equation in vacuum : Circular cylindrical cavity, axis along z-axis : // means tangent to wall S TM mode :
mj = jth zero of Jm(x) . resonant frequency with
Bessel Functions of Nonintegral Order Formally, gives only Jn of integral order. with However, the series expansion can be extended to J of nonintegral order : for 1, 2, 3, … Caution: are linearly independent.
Schlaefli Integral C encloses t = 0. n = integers For nonintegral , is multi-valued. Possible candidate for is Strategy for proving Show Fsatisfies Bessel eq. for J . Show for x 0. Mathematica
Consider any open contour C that doesn’t cross the branch cut
For C1 : this F is a solution of the Bessel eq. For C = spatial inversion of C , ( same as that for ; B.cut. on +axis ) . Set : Mathematica QED
2. Orthogonality where i.e., Z (k) is the eigenfunction, with eigenvalue k2, of the operator L . ( Sturm-Liouville eigenvalue probem ) • Helmholtz eq. in cylindrical coordinates • with • mn = nth root of Jm
is not self-adjoint is self-adjoint Jis an eigenfunction of L with eigenvalue k2. Lis Hermitian, i.e., , if the inner product is defined as
Orthogonal Sets Let Orthogonality : orthogonal set Let Orthogonality : orthogonal set
Mathematica Similarly : (see Ex.14.2.2)
Bessel Series : J( i / a ) For any well-behaved function f () with f (a) 0 : for any > 1 with
Bessel Series J( i / a ) For any well-behaved function f () with f (a) 0 : for any > 1 with
Example 14.2.1. 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 :
