8. Ch.3, part 2

Lecture 8 - Bessel Functions

The Laplace equation can also be separated in the circular cylindrical coordinates ρ, ϕ, z: it has solutions of the form R(ρ)Q(ϕ)Z(z). The one-dimensional equations in the ϕ, z coordinates are of the simplest type

\frac{d^{2}Z}{dz^{2}} − k^{2}Z = 0  (8.1)

\frac{d^{2}Q}{dϕ^{2}} + ν^{2}Q = 0  (8.2)

and all the complications are in the radial equation

\frac{d^{2}R}{dρ^{2}} + \frac{1}{ρ} \frac{dR}{dρ} + (k^{2} − \frac{ν^{2}}{ρ^{2}} ) R = 0  (8.3)

Putting x = kρ, this becomes the standard Bessel equation

\frac{d^{2}R}{dx^{2}} + \frac{1}{x} \frac{dR}{dx} + (1 − \frac{ν^{2}}{x^{2}} ) R = 0  (8.4)

In this chapter we deal only with solutions in the full azimuthal domain 0 ≤ ϕ ≤ 2π. Consequently the index ν takes on integer values and corresponds precisely to the index m of the spherical harmonics. However, we use the notation ν because non-integer values will be encountered later.

When k = 0 the solutions do not depend on z and the problem is in effect twodimensional. We already know that in this case the radial functions are simply ρ^{ν} and ρ^{−ν} for ν ≠ 0, and they are 1 and ln ρ for ν = 0.

For k ≠ 0 there are solutions that resemble sin(kx) and cos(kx), known as the Bessel functions J_{ν}(kρ), and solutions that resemble exp(ikx) and exp(−ikx), known as the Hankel functions H_{ν}^{(1)}(kρ) and H_{ν}^{(2)}(kρ). The Hankel functions will be used later, in radiation problems. Thus all we need for now are the Bessel functions for integer m.

8.1. Bessel functions of integer order

The functions J_{m}(kρ) are the analogues of the associated Legendre functions P_{l}^{m} (cos θ) , except that k is a continuous index (unlike l), if the domain of definition is 0 ≤ ρ ≤ ∞. As in the case of P_{l}^{m} and P_{l}^{−m}, negative values of m give nothing new and we can choose J_{−m}(kρ) = (−1)^{m}J_{m}(kρ). The functions J_{m} have series expansions and recursion relations (given later), but there is no Rodrigues formula for them, since they are not reducible to polynomials. Instead, there is the very useful integral representation

J_{m}(x) = \frac{1}{i^{m}} ∫_{0}^{2π} exp (ix cos φ − imφ) \frac{dφ}{2π}

(Jackson, Problem 3.14) that can be written in many other ways, including

J_{m}(x) = ∫_{0}^{π} cos (x cos φ − mφ) \frac{dφ}{π}

Closely related to these formulas (through Fourier’s theorem) is the generating function expansion

exp(ikρ cos φ) = ∑_{n=−∞}^{∞} i^{m} e^{imφ}J_{m}(kρ)

The useful lore about Bessel functions includes a large number of special integrals that we will discuss as needed. We note here only the orthogonality and normalization integral over the domain (0, ∞)

∫_{0}^{∞} J_{m}(kρ)J_{m}(k′ρ) ρ dρ = \frac{1}{k} δ(k − k′)

Successive J_{m} functions have interlacing zeros and maxima, like the Legendre functions. One should be familiar with the following plots, in which we note that J_{m}(x) behaves like x^{m} for small x. Thus it is regular at the origin.

Bessel functions of order 0,1,2 (above) and 5,6,7 (below)

For a given m^{2} there is also the “second solution” of the Bessel differential equation, which for small x behaves like x^{−m} for m > 0 and like ln x for m = 0. Its standard form is known as the Neumann function, or the Bessel function of the second kind, and is denoted with N_{m}(x) by some authors, Y_{m}(x) by others. It is discussed further here below. It is used in problems where ρ = 0 is not part of the domain.

We close this section by writing down the general “interior” solutions of the Laplace equations in cylindrical polar coordinates (they are the analog of the interior multipole fields of Lecture 7). They are, for k > 0,

e^{±kz} e^{±imϕ} J_{m}(kρ)

and represent standing waves in the radial direction, evanescent waves in the z direction. If instead of the Laplace equation we had solved the Helmholtz equation, ∇^{2}ψ = −(ω/c)^{2}ψ, the solutions would be

e^{±ik_{z}z} e^{±imϕ} J_{m}(k_{ρ}ρ)

with k_{z}^{2} + k_{ρ}^{2} = (ω/c)^{2}.

8.2. General Bessel and Hankel functions

Many properties of the Bessel functions hold for any value of the index ν and are sometimes easier to discuss for non-integer ν.

• Series expansions. For a given ν^{2}, the two solutions

J_{ν}(x) = (\frac{x}{2})^{ν} ∑_{n=0}^{∞} \frac{(−1)^{n}}{n!Γ(n + ν + 1)} (\frac{x}{2})^{2n}

J_{−ν}(x) = (\frac{x}{2})^{−ν} ∑_{n=0}^{∞} \frac{(−1)^{n}}{n!Γ(n − ν + 1)} (\frac{x}{2})^{2n}

are linearly independent, except when ν is an integer (note that the Γ function of a negative integer, or zero, is infinite). To avoid this difficulty, one defines the Neumann function

N_{ν}(x) = \frac{J_{ν}(x) cos νπ − J_{−ν}(x)}{sin νx}

which remains finite in the limit ν → integer.

• Hankel functions are defined as

H_{ν}^{(1)}(x) = J_{ν}(x) + iN_{ν}(x)
H_{ν}^{(2)}(x) = J_{ν}(x) − iN_{ν}(x)

Compare exp(±ix) = cos x ± i sin x.

• Recursion relations that hold for J_{ν} as well as for N_{ν}, H_{ν}^{(1)}, H_{ν}^{(2)} are

J_{ν−1} + J_{ν+1} = \frac{2ν}{x} J_{ν}

J_{ν−1} − J_{ν+1} = 2\frac{dJ_{ν}}{dx}

They are useful to compute integrals, but are numerically unstable as step-up relations.

• The behavior for small x can be read from the series (see J3.89-90). Note however the special case

N_{0}(x) → \frac{2}{π} (ln \frac{x}{2} + γ)

where γ ≃ 0.5772 is the Euler-Mascheroni constant.

• The asymptotic behavior for x >> ν is

J_{ν}(x) ∼ N_{ν−1}(x) ∼ \sqrt{\frac{2}{πx}} cos(x − \frac{νπ}{2} − \frac{π}{4} )

• Of special interest for fields confined inside a conducting cylinder are the zeroes of J_{m}(x). They are tabulated by Jackson. The orthogonality and normalization relations over a finite domain are also given. These relations are of interest to people who play (round) drums. They will also be useful to discuss the propagation of waves in cylindrical fibers and waveguides (when we get to it).

8.3. Modified Bessel functions

If we are going to make “virtual photons” out of the solutions of the wave equation

e^{±ik_{z}z} e^{±imϕ} J_{m}(k_{ρ}ρ)

all we have to require in the zero-frequency limit is that k_{z}^{2} + k_{ρ}^{2} = 0. Clearly, in addition to the option k_{z} = ik, k_{ρ} = k, we have the option k_{z} = k, k_{ρ} = ik, which gives the electrostatic solutions

e^{±ikz} e^{±imϕ} I_{m}(kρ)

where, by definition,

I_{m}(x) = i^{−m}J_{m}(ix)

is a modified Bessel function. I_{0}(x) resembles cosh x, I_{1}(x) resembles sinh x. Every I_{m} is well-behaved at the origin, but not at infinity: thus these functions are to be used for “interior” problems. The standard “second solution” is the modified Hankel function

K_{m}(x) = \frac{π}{2} i^{m+1}H_{m}^{(1)}(ix)

This blows up at the origin and vanishes at infinity: thus it is to be used for “exterior” problems. The above definitions are valid for any ν, not just for integer values m. The prefactors are chosen to make I_{ν}(x) and K_{ν}(x) real for real ν and x. See J3.102-104 for the precise behavior at x << 1 and x >> ν. Here are graphs of I_{0}(x), I_{1}(x), I_{2}(x) and of K_{0}(x), K_{1}(x), K_{2}(x) (solid, dashed, dotted)