8.3 Bessel Functions

In cylindrical polars, Laplace™s is

1 ‚2• ‚2•

1 ‚ ‚•

2

0= •= r + 2 2 + 2. (8.63)

r ‚r ‚r r ‚θ ‚z

If we set • = R(r)eimφ e±kx we ¬nd that R(r) obeys

d2 R 1 dR m2

2

+ k ’ 2 R = 0.

+ (8.64)

dr 2 r dr r

Now

d2 y ν2

1 dy

+ 1 ’ 2 y(x) = 0

+ (8.65)

dx2 x dx x

is Bessel™s equation and its solutions are Bessel functions of order ν. The

solutions for R will therefore be Bessel functions of order m, and with x

replaced by kr.

8.3.1 Cylindrical Bessel Functions

We now set about solving Bessel™s equation,

d2 y ν2

1 dy

+ 1 ’ 2 y(x) = 0.

+ (8.66)

dx2 x dx x

This has a regular singular point at the origin, and an irregular singular point

at in¬nity. We seek a series solution of the form

y = x» (1 + a1 x + a2 x2 + · · ·), (8.67)

and ¬nd from the indicial equation that » = ±ν. Setting » = ν and in-

serting the series into the equation, we ¬nd, with a conventional choice for

normalization, that

ν∞ (’1)n 2n

x x

def

y = Jν (x) = . (8.68)

2 n=0 n!(n + ν)! 2

Here (n + ν)! ≡ “(n + ν + 1).

8.3. BESSEL FUNCTIONS 231

If ν is an integer we ¬nd that J’n (x) = (’1)n Jn (x), so we have only

found one of the two independent solutions. Because of this, it is traditional

to de¬ne the Neumann function

Jν (x) cos νπ ’ J’ν (x)

Nν (x) = , (8.69)

sin νπ

as this remains an independent second solution even when ν becomes integral.

At short distance, and for ν not an integer

ν

x 1

+ ···,

Jν (x) =

2 “(ν + 1)

’ν

1 x

“(ν) + · · · .

Nν (x) = (8.70)

π 2

When ν tends to zero, we have

1

J0 (x) = 1 ’ x2 + · · ·

4

2

(ln x/2 + γ) + · · · ,

N0 (x) = (8.71)

π

where γ = ’“ (1) = .57721 . . . is the Euler-Mascheroni constant. For ¬xed

l, and x l we have the asymptotic expansions

2 1 1 1

Jν (x) ∼ cos(x ’ νπ ’ π) 1 + O , (8.72)

πx 2 4 x

2 1 1 1

Nν (x) ∼ sin(x ’ νπ ’ π) 1 + O . (8.73)

πx 2 4 x

It is therefore natural to de¬ne the Hankel functions

2 ix

(1)

Hν (x) = Jν (x) + iNν (x) ∼ e, (8.74)

πx

2 ’ix

(2)

Hν (x) = Jν (x) ’ iNν (x) ∼ e. (8.75)

πx

We will derive these asymptotic forms later.

232 CHAPTER 8. SPECIAL FUNCTIONS I

Generating Function

The two-dimensional wave equation

1 ‚2

2

’ 2 2 ¦(r, θ, t) = 0 (8.76)

c ‚t

has solutions

¦ = eiωt einθ Jn (kr), (8.77)

where k = |ω|/c. Equivalently, the two dimensional Helmholtz equation

2

+ k 2 )¦ = 0,

( (8.78)

has solutions einθ Jn (kr). It also has solutions with Jn (kr) replaced by Nn (kr),

but these are not ¬nite at the origin. Since the einθ Jn (kr) are the only

solutions that are ¬nite at the origin, any other ¬nite solution should be

expandable in terms of them. In particular, we should be able to expand a

plane wave solution in terms of them. For example,

eiky = eikr sin θ = an einθ Jn (kr). (8.79)

n

As we will see in a moment, the an ™s are all unity, so in fact

∞

ikr sin θ

einθ Jn (kr).

e = (8.80)

n=’∞

This generating function is the historical origin of the Bessel functions. They

were introduced by Bessel as a method of expressing the eccentric anomaly

of a planetary position as a Fourier sine series in the mean anomaly ” a

modern version of Hipparchus™ epicycles.

From the generating function we see that

1 2π

e’inθ+ix sin θ dθ.

Jn (x) = (8.81)

2π 0

Whenever you come across a formula like this, involving the Fourier integral

of the exponential of a trigonometric function, you are probably dealing with

a Bessel function.