2cosech(knm L) 2π a

Bnm =2 dθ U (r, θ)Jm (knm r) cos(mθ) rdr, m = 0,

πa [Jm (knm a)]2 0 0

(8.105)

and

1 2cosech(kn0 L) 2π a

Bn0 = dθ U (r, θ)J0 (kn0 r) rdr. (8.106)

2 πa2 [J0 (kn0 a)]2 0 0

Then we ¬t the boundary data expansion to the general solution, and so ¬nd

V (r, θ, z) = sinh(knm z)Jm (kmn r) [Anm sin(mθ) + Bnm cos(mθ)] . (8.107)

m,n

Hankel Transforms

When the radius, R, of the region in which we performing our eigenfunction

expansion becomes in¬nite, the eigenvalue spectrum will become continuous,

and the sum over the discrete kn Bessel-function zeros must be replaced by

an integral over k. By using the asymptotic approximation

2 1 1

Jn (kR) ∼ cos(kR ’ nπ ’ π), (8.108)

πkR 2 4

we may estimate the normalization integral as

R

R 2

Jm (kr) r dr ∼ + O(1). (8.109)

πk

0

We also ¬nd that the asymptotic density of Bessel zeros is

dn R

=. (8.110)

dk π

Putting these two results together shows that the continuous-spectrum or-

thogonality and completeness relations are

1

∞

δ(k ’ k ),

Jn (kr)Jn (k r) rdr = (8.111)

k

0

1

∞

δ(r ’ r ),

Jn (kr)Jn (kr ) kdk = (8.112)

r

0

respectively. These two equations establish that the Hankel transform (also

called the Fourier-Bessel transform) of a function f (r), which is de¬ned by

∞

F (k) = rdrJn (kr)f (r)r dr, (8.113)

0

has as its inverse ∞

f (r) = rdrJn (kr)F (k)k dk. (8.114)

0

240 CHAPTER 8. SPECIAL FUNCTIONS I

8.3.3 Modi¬ed Bessel Functions

The Bessel function Jn (kr) and the Neumann Nn (kr) function oscillate at

large distance, provided that k is real. When k is purely imaginary, it is

convenient to combine them so as to have functions that grow or decay ex-

ponentially. These are the modi¬ed Bessel functions.

We de¬ne

Iν (x) = i’ν Jν (ix), (8.115)

π

[I’ν (x) ’ Iν (x)].

Kν (x) = (8.116)

2 sin νπ

At short distance

ν

x

1

+ ···,

Iν (x) = (8.117)

2

“(ν + 1)

x ’ν

1

+ ···.

Kν (x) = “(ν) (8.118)

2 2

When ν becomes and integer we must take limits, and in particular

1

I0 (x) = 1 + x2 + · · · , (8.119)

4

K0 (x) = ’(ln x/2 + γ) + · · · . (8.120)

The large x asymptotic behaviour is

1

ex ,

Iν (x) ∼ √ x ’ ∞, (8.121)

2πx

π

Kν (x) ∼ √ e’x , x ’ ∞. (8.122)

2x

The factor of i’ν in the de¬nition of Iν (x) is to make Iν real.

From the expression for Jn (x) as an integral, we have

1 1

2π π

inθ x cos θ

cos(nθ)ex cos θ dθ

In (x) = e e dθ = (8.123)

2π π

0 0

for integer n. When n is not an integer we still have an expression for Iν (x)

as an integral, but now it is

1 sin νπ

π ∞

x cos θ

e’x cosh t’νt dt.

dθ ’

Iν (x) = cos(νθ)e (8.124)

π π

0 0

8.3. BESSEL FUNCTIONS 241

Here we need |arg x| < π/2 for the second integral to converge. The reason for

the “extra” in¬nite integral when ν in not an integer will not become obvious

until we learn how to use complex integral methods for solving di¬erential

equations. We will do this later. From the de¬nition of Kν (x) in terms of Iν

we ¬nd

∞

e’x cosh t cosh(νt) dt, |arg x| < π/2.

Kν (x) = (8.125)

0

Physics Illustration: Light propagation in optical ¬bres. Consider the propa-

gation of light of frequency ω0 down a straight section of optical ¬bre. Typical

¬bres are made of two materials. An outer layer, or cladding, with refractive

index n2 , and an inner core with refractive index n1 > n2 . The core of a ¬bre

used for communication is usually less than 10µm in diameter.

We will treat the light ¬eld E as a scalar. This is not a particularly good

approximation for real ¬bres, but the complications due the vector character

of the electromagnetic ¬eld are considerable. We suppose that E obeys

‚ 2 E ‚ 2 E ‚ 2 E n2 (x, y) ‚ 2 E

’

+ + = 0. (8.126)

‚x2 ‚y 2 ‚z 2 c2 ‚t2

Here n(x, y) is the refractive index of of the ¬bre, which is assumed to lie

along the z axis. We set

E(x, y, z, t) = ψ(x, y, z)eik0 z’iω0 t (8.127)

where k0 = ω0 /c. The amplitude ψ is a (relatively) slowly varying envelope

function. Plugging into the wave equation we ¬nd that

‚2ψ ‚2ψ ‚2ψ n2 (x, y) 2

‚ψ 2

ω0 ’ k0 ψ = 0.

+ 2 + 2 + 2ik0 + (8.128)

2 2