Bessel functions

SPECIAL FUNCTIONS
Using (18.65) and setting x = cos θ immediately gives a rearrangement of the required
result (18.69). Similarly, adding the plus and minus cases of result (18.68) gives
sin(n + 1)θ + sin(n − 1)θ = 2 sin nθ cos θ.
Dividing through on both sides by sin θ and using (18.66) yields (18.70). The recurrence relations (18.69) and (18.70) are extremely useful in the practical
computation of Chebyshev polynomials. For example, given the values of T0 (x)
and T1 (x) at some point x, the result (18.69) may be used iteratively to obtain
the value of any Tn (x) at that point; similarly, (18.70) may be used to calculate
the value of any Un (x) at some point x, given the values of U0 (x) and U1 (x) at
that point.
Further recurrence relations satisfied by the Chebyshev polynomials are
Tn (x) = Un (x) − xUn−1 (x),
(1 − x2 )Un (x) = xTn+1 (x) − Tn+2 (x),
(18.71)
(18.72)
which establish useful relationships between the two sets of polynomials Tn (x)
and Un (x). The relation (18.71) follows immediately from (18.68), whereas (18.72)
follows from (18.67), with n replaced by n + 1, on noting that sin2 θ = 1 − x2 .
Additional useful results concerning the derivatives of Chebyshev polynomials
may be obtained from (18.65) and (18.66), as illustrated in the following example.
Show that
Tn (x) = nUn−1 (x),
(1 − x2 )Un (x) = xUn (x) − (n + 1)Tn+1 (x).
These results are most easily derived from the expressions (18.65) and (18.66) by noting
that d/dx = (−1/ sin θ) d/dθ. Thus,
Tn (x) = −
Similarly, we find
1 d(cos nθ)
n sin nθ
=
= nUn−1 (x).
sin θ
dθ
sin θ
sin(n + 1)θ cos θ
(n + 1) cos(n + 1)θ
−
sin3 θ
sin2 θ
x Un (x) (n + 1)Tn+1 (x)
=
−
,
1 − x2
1 − x2
which rearranges immediately to yield the stated result. Un (x) = −
1 d
sin θ dθ
sin(n + 1)θ
sin θ
=
18.5 Bessel functions
Bessel’s equation has the form
x2 y + xy + (x2 − ν 2 )y = 0,
(18.73)
which has a regular singular point at x = 0 and an essential singularity at x = ∞.
The parameter ν is a given number, which we may take as ≥ 0 with no loss of
602
18.5 BESSEL FUNCTIONS
generality. The equation arises from physical situations similar to those involving
Legendre’s equation but when cylindrical, rather than spherical, polar coordinates
are employed. The variable x in Bessel’s equation is usually a multiple of a radial
distance and therefore ranges from 0 to ∞.
We shall seek solutions to Bessel’s equation in the form of infinite series. Writing
(18.73) in the standard form used in chapter 16, we have
1
ν2
(18.74)
y + y + 1 − 2 y = 0.
x
x
By inspection, x = 0 is a regular singular point; hence we try a solution of the
n
form y = xσ ∞
n=0 an x . Substituting this into (18.74) and multiplying the resulting
equation by x2−σ , we obtain
∞
∞
(σ + n)(σ + n − 1) + (σ + n) − ν 2 an xn +
an xn+2 = 0,
n=0
n=0
which simplifies to
∞
∞
(σ + n)2 − ν 2 an xn +
an xn+2 = 0.
n=0
n=0
0
Considering the coefficient of x , we obtain the indicial equation
σ 2 − ν 2 = 0,
and so σ = ±ν. For coefficients of higher powers of x we find
(σ + 1)2 − ν 2 a1 = 0,
(σ + n)2 − ν 2 an + an−2 = 0 for n ≥ 2.
(18.75)
(18.76)
Substituting σ = ±ν into (18.75) and (18.76), we obtain the recurrence relations
(1 ± 2ν)a1 = 0,
n(n ± 2ν)an + an−2 = 0
(18.77)
for n ≥ 2.
(18.78)
We consider now the form of the general solution to Bessel’s equation (18.73) for
two cases: the case for which ν is not an integer and that for which it is (including
zero).
18.5.1 Bessel functions for non-integer ν
If ν is a non-integer then, in general, the two roots of the indicial equation,
σ1 = ν and σ2 = −ν, will not differ by an integer, and we may obtain two linearly
independent solutions in the form of Frobenius series. Special considerations do
arise, however, when ν = m/2 for m = 1, 3, 5, . . . , and σ1 − σ2 = 2ν = m is an
(odd positive) integer. When this happens, we may always obtain a solution in
603
SPECIAL FUNCTIONS
the form of a Frobenius series corresponding to the larger root, σ1 = ν = m/2,
as described above. However, for the smaller root, σ2 = −ν = −m/2, we must
determine whether a second Frobenius series solution is possible by examining
the recurrence relation (18.78), which reads
n(n − m)an + an−2 = 0
for n ≥ 2.
Since m is an odd positive integer in this case, we can use this recurrence relation
(starting with a0 = 0) to calculate a2 , a4 , a6 , . . . in the knowledge that all these
terms will remain finite. It is possible in this case, therefore, to find a second
solution in the form of a Frobenius series, one that corresponds to the smaller
root σ2 .
Thus, in general, for non-integer ν we have from (18.77) and (18.78)
an
=
−
=
0
1
an−2
n(n ± 2ν)
for n = 2, 4, 6, . . . ,
for n = 1, 3, 5, . . . .
Setting a0 = 1 in each case, we obtain the two solutions
x4
x2
+
− ··· .
y±ν (x) = x±ν 1 −
2(2 ± 2ν) 2 × 4(2 ± 2ν)(4 ± 2ν)
It is customary, however, to set
a0 =
1
,
2±ν Γ(1 ± ν)
where Γ(x) is the gamma function, described in subsection 18.12.1; it may be
regarded as the generalisation of the factorial function to non-integer and/or
negative arguments.§ The two solutions of (18.73) are then written as Jν (x) and
J−ν (x), where
x ν 1 x 4
1
1
1 x 2
Jν (x) =
+
− ···
1−
Γ(ν + 1) 2
ν+1 2
(ν + 1)(ν + 2) 2! 2
∞
n
ν+2n
x
(−1)
=
;
(18.79)
n!Γ(ν + n + 1) 2
n=0
replacing ν by −ν gives J−ν (x). The functions Jν (x) and J−ν (x) are called Bessel
functions of the first kind, of order ν. Since the first term of each series is a
finite non-zero multiple of xν and x−ν , respectively, if ν is not an integer then
Jν (x) and J−ν (x) are linearly independent. This may be confirmed by calculating
the Wronskian of these two functions. Therefore, for non-integer ν the general
solution of Bessel’s equation (18.73) is given by
y(x) = c1 Jν (x) + c2 J−ν (x).
§
In particular, Γ(n + 1) = n! for n = 0, 1, 2,. . ., and Γ(n) is infinite if n is any integer ≤ 0.
604
(18.80)
18.5 BESSEL FUNCTIONS
We note that Bessel functions of half-integer order are expressible in closed form
in terms of trigonometric functions, as illustrated in the following example.
Find the general solution of
x2 y + xy + (x2 − 14 )y = 0.
This is Bessel’s equation with ν = 1/2, so from (18.80) the general solution is simply
y(x) = c1 J1/2 (x) + c2 J−1/2 (x).
However, Bessel functions of half-integral order can be expressed in terms of trigonometric
functions. To show this, we note from (18.79) that
J±1/2 (x) = x±1/2
∞
n=0
(−1)n x2n
.
22n±1/2 n!Γ(1 + n ± 12 )
Using the fact that Γ(x + 1) = xΓ(x) and Γ( 21 ) =
J1/2 (x) =
=
=
( 12 x)1/2
Γ( 23 )
( 12 x)1/2
√
( 12 ) π
( 12 x)1/2
√
( 12 ) π
√
π, we find that, for ν = 1/2,
( 1 x)5/2
( 1 x)9/2
− 2 5 + 2 7 − ···
1!Γ( 2 )
2!Γ( 2 )
( 12 x)5/2
( 12 x)9/2
−
√ +
√ − ···
1!( 32 )( 12 ) π
2!( 52 )( 32 )( 21 ) π
( 1 x)1/2 sin x
x2
x4
+
− · · · = 21 √
=
1−
3!
5!
(2) π x
2
sin x,
πx
whereas for ν = −1/2 we obtain
( 1 x)3/2
( 1 x)7/2
( 21 x)−1/2
− 2 3 + 2 5 − ···
1
Γ( 2 )
1!Γ( 2 )
2!Γ( 2 )
( 21 x)−1/2
x2
2
x4
1−
= √
+
− ··· =
cos x.
2!
4!
πx
π
J−1/2 (x) =
Therefore the general solution we require is
y(x) = c1 J1/2 (x) + c2 J−1/2 (x) = c1
2
sin x + c2
πx
2
cos x. πx
18.5.2 Bessel functions for integer ν
The definition of the Bessel function Jν (x) given in (18.79) is, of course, valid for
all values of ν, but, as we shall see, in the case of integer ν the general solution of
Bessel’s equation cannot be written in the form (18.80). Firstly, let us consider the
case ν = 0, so that the two solutions to the indicial equation are equal, and we
clearly obtain only one solution in the form of a Frobenius series. From (18.79),
605
SPECIAL FUNCTIONS
1.5
1
J0
J1
J2
0.5
2
4
6
8
10
x
−0.5
Figure 18.5 The first three integer-order Bessel functions of the first kind.
this is given by
J0 (x) =
∞
n=0
=1−
(−1)n x2n
22n n!Γ(1 + n)
x4
x6
x2
+ 2 2 − 2 2 2 + ··· .
22
2 4
2 46
In general, however, if ν is a positive integer then the solutions of the indicial
equation differ by an integer. For the larger root, σ1 = ν, we may find a solution
Jν (x), for ν = 1, 2, 3, . . . , in the form of the Frobenius series given by (18.79).
Graphs of J0 (x), J1 (x) and J2 (x) are plotted in figure 18.5 for real x. For the
smaller root, σ2 = −ν, however, the recurrence relation (18.78) becomes
n(n − m)an + an−2 = 0
for n ≥ 2,
where m = 2ν is now an even positive integer, i.e. m = 2, 4, 6, . . . . Starting with
a0 = 0 we may then calculate a2 , a4 , a6 , . . . , but we see that when n = m the
coefficient an is formally infinite, and the method fails to produce a second
solution in the form of a Frobenius series.
In fact, by replacing ν by −ν in the definition of Jν (x) given in (18.79), it can
be shown that, for integer ν,
J−ν (x) = (−1)ν Jν (x),
606
18.5 BESSEL FUNCTIONS
and hence that Jν (x) and J−ν (x) are linearly dependent. So, in this case, we cannot
write the general solution to Bessel’s equation in the form (18.80). One therefore
defines the function
Yν (x) =
Jν (x) cos νπ − J−ν (x)
,
sin νπ
(18.81)
which is called a Bessel function of the second kind of order ν (or, occasionally,
a Weber or Neumann function). As Bessel’s equation is linear, Yν (x) is clearly a
solution, since it is just the weighted sum of Bessel functions of the first kind.
Furthermore, for non-integer ν it is clear that Yν (x) is linearly independent of
Jν (x). It may also be shown that the Wronskian of Jν (x) and Yν (x) is non-zero
for all values of ν. Hence Jν (x) and Yν (x) always constitute a pair of independent
solutions.
If n is an integer, show that Yn+1/2 (x) = (−1)n+1 J−n−1/2 (x).
From (18.81), we have
Yn+1/2 (x) =
Jn+1/2 (x) cos(n + 12 )π − J−n−1/2 (x)
.
sin(n + 12 )π
If n is an integer, cos(n + 12 )π = 0 and sin(n + 12 )π = (−1)n , and so we immediately obtain
Yn+1/2 (x) = (−1)n+1 J−n−1/2 (x), as required. The expression (18.81) becomes an indeterminate form 0/0 when ν is an
integer, however. This is so because for integer ν we have cos νπ = (−1)ν and
J−ν (x) = (−1)ν Jν (x). Nevertheless, this indeterminate form can be evaluated using
l’Hôpital’s rule (see chapter 4). Therefore, for integer ν, we set
Jµ (x) cos µπ − J−µ (x)
Yν (x) = lim
,
(18.82)
µ→ν
sin µπ
which gives a linearly independent second solution for this case. Thus, we may
write the general solution of Bessel’s equation, valid for all ν, as
y(x) = c1 Jν (x) + c2 Yν (x).
(18.83)
The functions Y0 (x), Y1 (x) and Y2 (x) are plotted in figure 18.6
Finally, we note that, in some applications, it is convenient to work with
complex linear combinations of Bessel functions of the first and second kinds
given by
Hν(2) (x) = Jν (x) − iYν (x);
Hν(1) (x) = Jν (x) + iYν (x),
these are called, respectively, Hankel functions of the first and second kind of
order ν.
607
SPECIAL FUNCTIONS
1
Y0
0.5
2
Y1
Y2
4
6
8
10
x
−0.5
−1
Figure 18.6 The first three integer-order Bessel functions of the second kind.
18.5.3 Properties of Bessel functions Jν (x)
In physical applications, we often require that the solution is regular at x = 0,
but, from its definition (18.81) or (18.82), it is clear that Yν (x) is singular at
the origin, and so in such physical situations the coefficient c2 in (18.83) must
be set to zero; the solution is then simply some multiple of Jν (x). These Bessel
functions of the first kind have various useful properties that are worthy of
further discussion. Unless otherwise stated, the results presented in this section
apply to Bessel functions Jν (x) of integer and non-integer order.
Mutual orthogonality
In section 17.4, we noted that Bessel’s equation (18.73) could be put into conventional Sturm–Liouville form with p = x, q = −ν 2 /x, λ = α2 and ρ = x,
provided αx is the argument of y. From the form of p, we see that there is no
natural interval over which one would expect the solutions of Bessel’s equation
corresponding to different eigenvalues λ (but fixed ν) to be automatically orthogonal. Nevertheless, provided the Bessel functions satisfied appropriate boundary
conditions, we would expect them to obey an orthogonality relationship over
some interval [a, b] of the form
b
xJν (αx)Jν (βx) dx = 0
for α = β.
(18.84)
a
608
18.5 BESSEL FUNCTIONS
To determine the required boundary conditions for this result to hold, let us
consider the functions f(x) = Jν (αx) and g(x) = Jν (βx), which, as will be proved
below, respectively satisfy the equations
x2 f + xf + (α2 x2 − ν 2 )f = 0,
(18.85)
x2 g + xg + (β 2 x2 − ν 2 )g = 0.
(18.86)
Show that f(x) = Jν (αx) satisfies (18.85).
If f(x) = Jν (αx) and we write w = αx, then
df
d2 Jν (w)
dJν (w)
d2 f
= α2
.
=α
and
dx
dw
dx2
dw 2
When these expressions are substituted into (18.85), its LHS becomes
dJν (w)
d2 Jν (w)
+ xα
+(α2 x2 − ν 2 )Jν (w)
dw 2
dw
dJν (w)
d2 Jν (w)
= w2
+w
+ (w 2 − ν 2 )Jν (w).
dw 2
dw
But, from Bessel’s equation itself, this final expression is equal to zero, thus verifying that
f(x) does satisfy (18.85). x 2 α2
Now multiplying (18.86) by f(x) and (18.85) by g(x) and subtracting them gives
d
[x(fg − gf )] = (α2 − β 2 )xfg,
dx
where we have used the fact that
d
[x(fg − gf )] = x(fg − gf ) + (fg − gf ).
dx
By integrating (18.87) over any given range x = a to x = b, we obtain
b
b
1
xf(x)g (x) − xg(x)f (x) ,
xf(x)g(x) dx = 2
2
a
α −β
a
(18.87)
which, on setting f(x) = Jν (αx) and g(x) = Jν (βx), becomes
b
b
1
βxJ
xJν (αx)Jν (βx) dx = 2
(αx)J
(βx)
−
αxJ
(βx)J
(αx)
.
ν
ν
ν
ν
a
α − β2
a
(18.88)
If α = β, and the interval [a, b] is such that the expression on the RHS of (18.88)
equals zero, then we obtain the orthogonality condition (18.84). This happens, for
example, if Jν (αx) and Jν (βx) vanish at x = a and x = b, or if Jν (αx) and Jν (βx)
vanish at x = a and x = b, or for many more general conditions. It should be
noted that the boundary term is automatically zero at the point x = 0, as one
might expect from the fact that the Sturm–Liouville form of Bessel’s equation
has p(x) = x.
If α = β, the RHS of (18.88) takes the indeterminant form 0/0. This may be
609
SPECIAL FUNCTIONS
evaluated using l’Hôpital’s rule, or alternatively we may calculate the relevant
integral directly.
Evaluate the integral
b
Jν2 (αx)x dx.
a
Ignoring the integration limits for the moment,
1
Jν2 (αx)x dx = 2
Jν2 (u)u du,
α
where u = αx. Integrating by parts yields
I = Jν2 (u)u du = 12 u2 Jν2 (u) − Jν (u)Jν (u)u2 du.
Now Bessel’s equation (18.73) can be rearranged as
u2 Jν (u) = ν 2 Jν (u) − uJν (u) − u2 Jν (u),
which, on substitution into the expression for I, gives
I = 12 u2 Jν2 (u) − Jν (u)[ν 2 Jν (u) − uJν (u) − u2 Jν (u)] du
= 12 u2 Jν2 (u) − 12 ν 2 Jν2 (u) + 12 u2 [Jν (u)]2 + c.
Since u = αx, the required integral is given by
b
b
1
ν2
x2 − 2 Jν2 (αx) + x2 [Jν (αx)]2 ,
Jν2 (αx)x dx =
2
α
a
a
(18.89)
which gives the normalisation condition for Bessel functions of the first kind. Since the Bessel functions Jν (x) possess the orthogonality property (18.88), we
may expand any reasonable function f(x), i.e. one obeying the Dirichlet conditions
discussed in chapter 12, in the interval 0 ≤ x ≤ b as a sum of Bessel functions of
a given (non-negative) order ν,
f(x) =
∞
cn Jν (αn x),
(18.90)
n=0
provided that the αn are chosen such that Jν (αn b) = 0. The coefficients cn are then
given by
b
2
f(x)Jν (αn x)x dx.
(18.91)
cn = 2 2
b Jν+1 (αn b) 0
The interval is taken to be 0 ≤ x ≤ b, as then one need only ensure that the
appropriate boundary condition is satisfied at x = b, since the boundary condition
at x = 0 is met automatically.
610
18.5 BESSEL FUNCTIONS
Prove the expression (18.91).
If we multiply (18.90) by xJν (αm x) and integrate from x = 0 to x = b then we obtain
b
b
∞
xJν (αm x)f(x) dx =
cn
xJν (αm x)Jν (αn x) dx
0
0
n=0
b
Jν2 (αm x)x dx
= cm
0
2
= 12 cm b2 J ν (αm b) = 12 cm b2 Jν+1
(αm b),
2
where in the last two lines we have used (18.88) with αm = α = β = αn , (18.89), the fact
that Jν (αm b) = 0 and (18.95), which is proved below. Recurrence relations
The recurrence relations enjoyed by Bessel functions of the first kind, Jν (x), can
be derived directly from the power series definition (18.79).
Prove the recurrence relation
d ν
[x Jν (x)] = xν Jν−1 (x).
dx
(18.92)
From the power series definition (18.79) of Jν (x) we obtain
∞
(−1)n x2ν+2n
d ν
d [x Jν (x)] =
ν+2n
dx
dx n=0 2
n!Γ(ν + n + 1)
=
∞
n=0
= xν
(−1)n x2ν+2n−1
2ν+2n−1 n!Γ(ν + n)
∞
n=0
(−1)n x(ν−1)+2n
= xν Jν−1 (x). 2(ν−1)+2n n!Γ((ν − 1) + n + 1)
It may similarly be shown that
d −ν
[x Jν (x)] = −x−ν Jν+1 (x).
(18.93)
dx
From (18.92) and (18.93) the remaining recurrence relations may be derived.
Expanding out the derivative on the LHS of (18.92) and dividing through by
xν−1 , we obtain the relation
xJν (x) + νJν (x) = xJν−1 (x).
(18.94)
Similarly, by expanding out the derivative on the LHS of (18.93), and multiplying
through by xν+1 , we find
xJν (x) − νJν (x) = −xJν+1 (x).
(18.95)
Adding (18.94) and (18.95) and dividing through by x gives
Jν−1 (x) − Jν+1 (x) = 2Jν (x).
611
(18.96)
SPECIAL FUNCTIONS
Finally, subtracting (18.95) from (18.94) and dividing by x gives
Jν−1 (x) + Jν+1 (x) =
2ν
Jν (x).
x
(18.97)
Given that J1/2 (x) = (2/πx)1/2 sin x and that J−1/2 (x) = (2/πx)1/2 cos x, express J3/2 (x)
and J−3/2 (x) in terms of trigonometric functions.
From (18.95) we have
1
(x)
J1/2 (x) − J1/2
2x
1/2
1/2
1/2
1
1
2
2
2
sin x −
cos x +
sin x
=
2x πx
πx
2x πx
1/2 1
2
=
sin x − cos x .
πx
x
J3/2 (x) =
Similarly, from (18.94) we have
1
(x)
J−1/2 (x) + J−1/2
2x
1/2
1/2
1/2
1
1
2
2
2
cos x −
sin x −
cos x
=−
2x πx
πx
2x πx
1/2 1
2
− cos x − sin x .
=
πx
x
J−3/2 (x) = −
We see that, by repeated use of these recurrence relations, all Bessel functions Jν (x) of halfinteger order may be expressed in terms of trigonometric functions. From their definition
(18.81), Bessel functions of the second kind, Yν (x), of half-integer order can be similarly
expressed. Finally, we note that the relations (18.92) and (18.93) may be rewritten in
integral form as
xν Jν−1 (x) dx = xν Jν (x),
x−ν Jν+1 (x) dx = −x−ν Jν (x).
If ν is an integer, the recurrence relations of this section may be proved using
the generating function for Bessel functions discussed below. It may be shown
that Bessel functions of the second kind, Yν (x), also satisfy the recurrence relations
derived above.
Generating function
The Bessel functions Jν (x), where ν = n is an integer, can be described by a
generating function in a way similar to that discussed for Legendre polynomials
612
18.5 BESSEL FUNCTIONS
in subsection 18.1.2. The generating function for Bessel functions of integer order
is given by
∞
x
1
Jn (x)hn .
(18.98)
h−
=
G(x, h) = exp
2
h
n=−∞
By expanding the exponential as a power series, it is straightfoward to verify that
the functions Jn (x) defined by (18.98) are indeed Bessel functions of the first kind,
as given by (18.79).
The generating function (18.98) is useful for finding, for Bessel functions of
integer order, properties that can often be extended to the non-integer case. In
particular, the Bessel function recurrence relations may be derived.
Use the generating function to prove, for integer ν, the recurrence relation (18.97), i.e.
Jν−1 (x) + Jν+1 (x) =
2ν
Jν (x).
x
Differentiating G(x, h) with respect to h we obtain
∞
1
∂G(x, h)
x
1 + 2 G(x, h) =
nJn (x)hn−1 ,
=
∂h
2
h
n=−∞
which can be written using (18.98) again as
∞
∞
1 x
1+ 2
Jn (x)hn =
nJn (x)hn−1 .
2
h n=−∞
n=−∞
Equating coefficients of hn we obtain
x
[Jn (x) + Jn+2 (x)] = (n + 1)Jn+1 (x),
2
which, on replacing n by ν − 1, gives the required recurrence relation. Integral representations
The generating function (18.98) is also useful for deriving integral representations
of Bessel functions of integer order.
Show that for integer n the Bessel function Jn (x) is given by
1 π
cos(nθ − x sin θ) dθ.
Jn (x) =
π 0
(18.99)
By expanding out the cosine term in the integrand in (18.99) we obtain the integral
1 π
[cos(x sin θ) cos nθ + sin(x sin θ) sin nθ] dθ.
I=
(18.100)
π 0
Now, we may express cos(x sin θ) and sin(x sin θ) in terms of Bessel functions by setting
h = exp iθ in (18.98) to give
∞
x
exp (exp iθ − exp(−iθ)) = exp (ix sin θ) =
Jm (x) exp imθ.
2
m=−∞
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