Spherical Bessel functions

SPECIAL FUNCTIONS
Using de Moivre’s theorem, exp iθ = cos θ + i sin θ, we then obtain
exp (ix sin θ) = cos(x sin θ) + i sin(x sin θ) =
∞
Jm (x)(cos mθ + i sin mθ).
m=−∞
Equating the real and imaginary parts of this expression gives
cos(x sin θ) =
sin(x sin θ) =
∞
m=−∞
∞
Jm (x) cos mθ,
Jm (x) sin mθ.
m=−∞
Substituting these expressions into (18.100) then yields
∞ π
1 [Jm (x) cos mθ cos nθ + Jm (x) sin mθ sin nθ] dθ.
I=
π m=−∞ 0
However, using the orthogonality of the trigonometric functions [ see equations (12.1)–
(12.3) ], we obtain
1π
I=
[Jn (x) + Jn (x)] = Jn (x),
π2
which proves the integral representation (18.99). Finally, we mention the special case of the integral representation (18.99) for
n = 0:
2π
1 π
1
J0 (x) =
cos(x sin θ) dθ =
cos(x sin θ) dθ,
π 0
2π 0
since cos(x sin θ) repeats itself in the range θ = π to θ = 2π. However, sin(x sin θ)
changes sign in this range and so
2π
1
sin(x sin θ) dθ = 0.
2π 0
Using de Moivre’s theorem, we can therefore write
2π
2π
1
1
J0 (x) =
exp(ix sin θ) dθ =
exp(ix cos θ) dθ.
2π 0
2π 0
There are in fact many other integral representations of Bessel functions; they
can be derived from those given.
18.6 Spherical Bessel functions
When obtaining solutions of Helmholtz’ equation (∇2 + k 2 )u = 0 in spherical
polar coordinates (see section 21.3.2), one finds that, for solutions that are finite
on the polar axis, the radial part R(r) of the solution must satisfy the equation
r 2 R + 2rR + [k 2 r 2 − ( + 1)]R = 0,
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(18.101)
18.6 SPHERICAL BESSEL FUNCTIONS
where is an integer. This equation looks very much like Bessel’s equation and
can in fact be reduced to it by writing R(r) = r −1/2 S(r), in which case S(r) then
satisfies
2 S = 0.
r 2 S + rS + k 2 r 2 − + 12
On making the change of variable x = kr and letting y(x) = S(kr), we obtain
x2 y + xy + [x2 − ( + 12 )2 ]y = 0,
where the primes now denote d/dx. This is Bessel’s equation of order + 12
and has as its solutions y(x) = J+1/2 (x) and Y+1/2 (x). The general solution of
(18.101) can therefore be written
R(r) = r −1/2 [c1 J+1/2 (kr) + c2 Y+1/2 (kr)],
where c1 and c2 are constants that may be determined from the boundary
conditions on the solution. In particular, for solutions that are finite at the origin
we require c2 = 0.
The functions x−1/2 J+1/2 (x) and x−1/2 Y+1/2 (x), when suitably normalised, are
called spherical Bessel functions of the first and second kind, respectively, and are
denoted as follows:
j (x) =
n (x) =
π
J+1/2 (x),
2x
π
Y+1/2 (x).
2x
(18.102)
(18.103)
For integer , we also note that Y+1/2 (x) = (−1)+1 J−−1/2 (x), as discussed in
section 18.5.2. Moreover, in section 18.5.1, we noted that Bessel functions of the
first kind, Jν (x), of half-integer order are expressible in closed form in terms of
trigonometric functions. Thus, all spherical Bessel functions of both the first and
second kinds may be expressed in such a form. In particular, using the results of
the worked example in section 18.5.1, we find that
sin x
,
x
cos x
n0 (x) = −
.
x
j0 (x) =
(18.104)
(18.105)
Expressions for higher-order spherical Bessel functions are most easily obtained
by using the recurrence relations for Bessel functions.
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