Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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286 8 Legendre and Bessel Functions
1 0.5 0 0.5 1
x
1.5
1
0.5
0
0.5
1
1.5
2
Q
n
x
Figure 8.2. Legendre functions of second kind.
nor space permits full development here of the theory of hypergeometric functions, which
can be found in more advanced texts, it is still of interest to express the Legendre functions
in terms of hypergeometric functions.
Consider the variable transformation
t
1
2
1 zu
'
t2y
'
z ,u
''
t4y
''
z ,t1 t
1
4
1 z
2
(8.131)
such that
1 z
2
y
''
z
2ΓΑΒ11 z
y
'
zΑΒyz0 (8.132)
By choosing
ΑΒ1 2Γ0 (8.133)
ΑΒ1 2 Γ1 , ΑΝ, ΒΝ1 (8.134)
ΑΒ ΝΝ 1 (8.135)
we obtain Legendre’s equation
1 z
2
y
''
z2zy
'
zΝΝ1yz0 (8.136)
and identify
z 1 < 2 P
Ν
zF
Ν, Ν1, 1,
1 z
2
(8.137)
P
Ν,Μ
z
1
A1 Μ
z 1
z 1
Μ/ 2
F
Ν, Ν1, 1 Μ,
1 z
2
(8.138)
8.2 Legendre Functions 287
as the Legendre function of the first kind in term of a Taylor series around z 1 with the
conventional normalization P
Ν
11. Alternatively, the substitutions
t z
2
, Α
Ν2
2
, Β
Ν1
2
, Γ
Ν3
2
(8.139)
also transform the hypergeometric equation into Legendre’s equation, permitting the
Legendre functions of the second kind to be identified as
z > 1 Q
Ν
z
Π
2z
Ν1
AΝ 1
A
Ν
3
2
F
Ν2
2
,
Ν1
2
,
Ν3
2
,z
2
(8.140)
Q
Ν,Μ
z
Π
2
Ν1
z
2
1
Μ/ 2
z
ΝΜ1
AΝΜ1
A
Ν
3
2
ΜΠ
F
ΝΜ2
2
,
ΝΜ1
2
,
Ν3
2
,z
2
(8.141)
when Ν is not a negative integer. The first factor is made single-valued by a cut on (, 1)
and the normalization is conventional. Definitions for larger regions can be obtained by
analytic continuation, but we refer the reader to standard references for further details.
8.2.11 Analytic Structure of Legendre Functions
The most general form of the associated Legendre equation is
1 z
2
y
''
z2zy
'
z
ΝΝ 1
Μ
2
1 z
2
yz0
yzAP
Ν,Μ
zBQ
Ν,Μ
z (8.142)
where z, Ν, Μ are permitted to be complex. Most of our results for Legendre functions of
real arguments in the range 1 <x<1 can be extended to the unit disk z < 1simply
by replacing x ! z in series representations, recursion relations, and other formulas even
if the degree Ν or order Μ is permitted to be complex. However, for arbitrary z one must
pay close attention to the choice of branch cuts. The most common, although not unique,
choices are listed in Table 8.2. Each is then an entire function of z in the cut plane, and
customary values on the real axis for 1 <x<1 can be defined using suitable averages of
values above and below the cut where Ε!0
.
Table 8.2. Branch cuts for Legendre functions.
Function Cut Value for 1 <x<1
P
Ν
z, 1 P
Ν
x
Q
Ν
z, 1 Q
Ν
xQ
Ν
x ΕQ
Ν
x Ε/ 2
P
Ν,Μ
z, 1 P
Ν,Μ
xExpΠΜ/ 2P
Ν,Μ
x Ε
Q
Ν,Μ
z, 1 Q
Ν,Μ
xExpΠΜExpΠΜ/ 2Q
Ν,Μ
x ΕExpΠΜ/ 2Q
Ν,Μ
x Ε/ 2
288 8 Legendre and Bessel Functions
If we define
f
Ν,Μ
zAP
Ν,Μ
zBQ
Ν,Μ
z (8.143)
where A, B are constant coefficients independent of Ν, Μ,z, we find that the recursion
relations
f
Ν,Μ2
Μ1
2z
z
2
1
1/ 2
f
Ν,Μ1
ΝΝ 1ΜΜ1
f
Ν,Μ
0 (8.144)
zf
Ν,Μ
ΝΜ1
z
2
1
1/ 2
f
Ν,Μ1
f
Ν1,Μ
0 (8.145)
f
Ν1,Μ1
2Ν1
z
2
1
1/ 2
f
Ν,Μ
f
Ν1,Μ1
0 (8.146)
Ν Μ 1 f
Ν1,Μ
2Ν1zf
Ν,Μ
ΝΜf
Ν1,Μ
0 (8.147)
appear slightly different but that the previous results for z ! x with 1 <x<1 will be
recovered if z
2
1
1/ 2
is also handled with a cut on (, 1) in a consistent manner.
Similarly, the Rodrigues formulas now take the form
P
Ν,m
zz
2
1
m/2
z
m
P
Ν
zP
Ν,m
x
m
1 x
2
m/2
x
m
P
Ν
x (8.148)
Q
Ν,m
zz
2
1
m/2
z
m
Q
Ν
zQ
Ν,m
x
m
1 x
2
m/2
x
m
Q
Ν
x (8.149)
for nonnegative integer m. These results offer another justification for retention of the
m
phase when the argument x CosΘ is in the range (1, 1).
Generalizing the Schläfli integral representation of P
n
z, one can show that
P
Ν
z
2
Ν
2Π
C
1
t
2
1
Ν
t z
Ν1
t (8.150)
Q
Ν
z
2
Ν
4 SinΝΠ
C
2
t
2
1
Ν
z t
Ν1
t (8.151)
provide solutions to Legendre’s differential equation for arbitrary n !Ν, but the contours
for noninteger Ν must be chosen carefully because the integrands have branch points at t
1, z, 1. Suitable contours are sketched in Fig. 8.3. Legendre functions of the first kind
are obtained using contour C
1
that encloses both t z and t 1 with one cut below the
real axis for t 1 and a second between t 1 and t z. After integration with respect
to t, the cut between t 1 and z becomes irrelevant but P
Ν
z is left with a branch point at
z 1 for noninteger Ν and a cut below the negative real axis from there to . With these
choices we find that P
Ν
11 for any Ν. Alternatively, the bow tie contour C
2
provides a
related integral representation for Q
Ν
z .ForC
2
, z must be outside both loops so that the
phase changes, incurred by circumnavigating the branch points, cancel. A more detailed
analysis shows that a single-valued definition of Q
Ν
requires a cut from z 1toz .
Similar integral representations can also be constructed for associated Legendre functions,
but we will be content to stop here.
8.3 Bessel Functions 289
Ret
Imt
1
z
P
Ν
z
Ret
Imt
z
1 1
Q
Ν
z
Figure 8.3. Contours for integral representations of Legendre functions.
Finally, one can also show that, for fixed z, P
Ν
is an entire function of Ν while Q
Ν
is a meromorphic function of Ν with simple poles at negative integers. This property of
Legendre functions is useful in quantum scattering theory where significant insights can
be obtained by treating the orbital angular momentum as a complex variable. More detailed
development of the properties of Legendre functions of the second kind can be obtained,
when needed, from standard references. These developments are omitted here because we
will not find much use for these functions in the remainder of the course. Suffice it to say
that most properties can be derived from the Schläfli integral representation.
8.3 Bessel Functions
8.3.1 Cylindrical
Variants of Bessel’s differential equation arise frequently when the method of separation
of variables is applied in cylindrical or spherical coordinates to partial differential equa-
tions based upon the Laplacian operator. However, it is instructive to begin our study of
these functions using the generating function instead of the differential equation, similar to
the approach used for Legendre functions. The generating function for cylindrical Bessel
functions of the first kind, J
n
x, takes the form of a Laurent expansion with respect to t
gt,xExp
x
2
t t
1
n
J
n
xt
n
(8.152)
whose coefficients are the desired functions of x. Bessel functions with negative n are
related to those with positive n using the symmetry
gt
1
,xgt,xJ
n
x
n
J
n
x (8.153)
Similarly, we obtain the reflection symmetry
gt,xgt,xJ
n
x
n
J
n
x (8.154)
such that
J
n
xJ
n
x (8.155)
for integer n.
290 8 Legendre and Bessel Functions
Some of the basic recursion relations can be obtained, as before, by differentiating
gt,x with respect to either variable. First consider
(gt,x
(t
x
2t
2
t
2
1 Exp
x
2
t t
1
n
nJ
n
xt
n1
(8.156)
and substitute the generating function to write
x
2t
2
t
2
1
n
J
n
xt
n
n
nJ
n
xt
n1
n
x
2
J
n
x
2
J
n2
n 1J
n1
t
n
0 (8.157)
If this series is to vanish for arbitrary t, the coefficients must vanish separately for each
term. Adjusting the indexing, we obtain the three-term recursion relation
2n
x
J
n
xJ
n1
xJ
n1
x (8.158)
Similarly, differentiation with respect to x
(gt,x
(x
1
2
t t
1
n
J
n
xt
n
n
J
'
n
xt
n
(8.159)
provides a recursion relation
2J
'
n
xJ
n1
xJ
n1
x (8.160)
for the derivative, almost by inspection. Manipulation of these recursion relations provides
several additional relationships:
J
n1
x
n
x
J
n
x J
'
n
x (8.161)
x
x
n
J
n
x x
n
J
n1
x (8.162)
A power series for J
n
is obtained by factoring the generating function and expanding
each factor according to
gt,xExp
xt
2
Exp
x
2t
3
4
4
4
4
4
5
n0
1
n!
xt
2
n
6
7
7
7
7
7
8
3
4
4
4
4
4
5
m0
1
m!
x
2t
m
6
7
7
7
7
7
8
n,m0
m
n!m!
x
2
nm
t
nm
(8.163)
The coefficients for each nonnegative power of t are identified as
n 2 0 J
n
x
m0
m
n m!m!
x
2
n2m
(8.164)
8.3 Bessel Functions 291
10 5 0 5 10
x
0.5
0.25
0
0.25
0.5
0.75
1
J
n
x
Figure 8.4. Bessel functions of first kind.
or
n 2 0 J
n
x
x
2
n
m0
1
n m!m!
x
2
4
m
(8.165)
where the leading power has been factored out and the series written as an even function
of x. Similarly, for negative powers we obtain
J
n
x
m0
m
m n!m!
x
2
2mn
(8.166)
However, because the coefficients vanish unless m 2 n,weusem ! n k to write
J
n
x
k0
nk
n k!m!
x
2
n2k
J
n
x
n
J
n
x (8.167)
and again conclude that changing the sign of the index does not yield an independent
function. One can show that the power series is convergent for all x,evenifitisinefficient
for large x.
Figure 8.4 displays the first few Bessel functions of the first kind. The functions oscil-
late with slowly decreasing amplitudes. Their roots are interleaved, as expected for solu-
tions of a Sturm–Liouville system.
Notice that gt,z is an analytic function of either variable, except for the essential
singularity at t 0. Thus,
J
n
z
m0
m
n m!m!
z
2
n2m
(8.168)
292 8 Legendre and Bessel Functions
is an analytic function whose radius of convergence is infinite. We can generalize this
series
J
Ν
z
z
2
Ν
m0
1
AΝ m 1Am 1
z
2
4
m
(8.169)
to arbitrary Ν using a branch point at the origin for noninteger Ν and intepreting the factori-
als as gamma functions. The cut is normally taken below the negative real axis. However,
the symmetry between positive and negative integer order n does not apply for noninte-
ger Ν. Near the origin one finds
J
Ν
z*AΝ1
1
z
2
Ν
(8.170)
provided that Ν is not a negative integer. Recognizing that the first n 1 coefficients of the
power series vanish when Νn, we find
J
n
z
n
J
n
z*An 1
1
z
2
n
(8.171)
as expected.
It is now a simple matter to verify that the basic recursion relations
2Ν
z
J
Ν
zJ
Ν1
zJ
Ν1
z (8.172)
2J
'
Ν
zJ
Ν1
zJ
Ν1
z (8.173)
continue to apply for noninteger Ν and complex z. The differential equation satisfied by
J
Ν
z can be deduced from the recursion relations (or from the series). Suppose that f
Ν
z
is a family of functions that satisfy the basic recursion relations
2Ν f
Ν
zz f
Ν1
z f
Ν1
z (8.174)
2 f
'
Ν
z f
Ν1
z f
Ν1
z (8.175)
but are not necessarily limited to cylindrical Bessel functions of the first kind. From these
one deduces
f
Ν1
z
Ν
z
f
Ν
z f
'
Ν
z (8.176)
f
Ν1
z
Ν
z
f
Ν
z f
'
Ν
z (8.177)
such that
zf
'
Ν
zzf
Ν1
zΝf
Ν
zzf
''
Ν
zzf
'
Ν1
z f
Ν1
zΝ1 f
'
Ν
z0 (8.178)
or
z
2
f
''
Ν
zz
2
f
'
Ν1
zzf
Ν1
zΝ1zf
'
Ν
z0 (8.179)
8.3 Bessel Functions 293
Substituting
zf
'
Ν1
zΝ1 f
Ν1
zzf
Ν
z (8.180)
zf
Ν1
zΝf
Ν
zzf
'
Ν
z (8.181)
such that
z
2
f
'
Ν1
z
ΝΝ 1z
2
f
Ν
zΝ1zf
'
Ν
z (8.182)
one obtains
z
2
f
''
Ν
z
ΝΝ 1z
2
f
Ν
zΝ1zf
'
Ν
z
Νf
Ν
zzf
'
Ν
z
Ν1zf
'
Ν
z0 (8.183)
Therefore, we finally obtain Bessel’s differential equation
z
2
f
''
Ν
zzf
'
Ν
zz
2
Ν
2
f
Ν
z0 (8.184)
for which J
Ν
z are solutions that are regular at the origin for integer Ν. Since this is a
second-order differential equation, there must exist a second linearly independent set of
solutions that also satisfy the same recursion relations.
It is important to recognize that the same recursion relations apply to any solution of
Bessel’s equation or to any linear combination of solutions with the same Ν, provided that
the coefficients are independent of both Ν and z. Thus, the recursion relations
f
Ν1
z f
Ν1
z
2Ν
z
f
Ν
z (8.185)
f
Ν1
z f
Ν1
z2 f
'
Ν
z (8.186)
z
z
Ν
f
Ν
z
z
Ν
f
Ν1
z (8.187)
are general properties of cylindrical Bessel functions. (We leave it as an exercise to the
reader to derive the third formula from the first two.)
If we divide by z, Bessel’s equation appears to be self-adjoint with respect to the weight
function z provided that we use appropriate boundary conditions. If we assume that Ν20
(the usual situation), the requirement that the solution be finite at the origin selects J
Ν
while
requiring either a node or a vanishing derivative at a fixed radius R provides an eigenvalue
equation for either Dirichlet or Neumann boundary conditions. Thus, it is useful to change
variables such that
1
Ξ
Ξ
Ξ f
'
Ξ
k
2
Ν
2
Ξ
2
f Ξ 0 ,f0 finite f Ξ - J
Ν
kΞ (8.188)
where the boundary conditions
Dirichlet: f R0 J
Ν
k
Ν,n
Ξ 0 (8.189)
Neumann: f
'
R0 J
'
Ν
k
Ν,n
Ξ 0 (8.190)
yield sets of eigenfunctions indexed by the positive integer n. It is now a simple matter,
following the procedures of the Sturm–Liouville chapter, to demonstrate orthogonality
294 8 Legendre and Bessel Functions
between two solutions with the same Ν but different eigenvalues. Therefore, we obtain
eigenfunction expansions of the form
FΞ
n
A
n
J
Ν
k
Ν,n
Ξ
(8.191)
with the aid of the orthonormality relations
J
Ν
k
Ν,n
R0
R
0
J
Ν
k
Ν,m
ΞJ
Ν
k
Ν,n
ΞΞ Ξ
R
2
2
J
Ν1
k
Ν,n
R
2
∆
m,n
(8.192)
J
'
Ν
k
Ν,n
R0
R
0
J
Ν
k
Ν,m
ΞJ
Ν
k
Ν,n
ΞΞ Ξ
R
2
2
3
4
4
4
4
5
1
Ν
k
Ν,n
R
2
6
7
7
7
7
8
J
Ν
k
Ν,n
R
2
∆
m,n
(8.193)
that are developed in the exercises. The appropriate choice of basis depends upon the
boundary conditions. Note that summation is performed over n for fixed Ν20. The appro-
priate value of Ν is determined by other aspects of the problem, such as angular momentum.
An important special case is the limit R !for which the spacing between eigenvalues
becomes infinitesimal. With careful attention to the limit, one can prove that
0
J
Ν
kΞJ
Ν
k
'
ΞΞ Ξ
∆k k
'
kk
'
(8.194)
0
J
Ν
kΞJ
Ν
kΞ
'
k k
∆Ξ Ξ
'
ΞΞ
'
(8.195)
for Ν20. The right-hand side is the appropriate delta function for cylindrical coordinates
with a linear weight factor; here the denominator is written in a symmetric form and bal-
ances the weight factor such that
0
Ξ Ξ f Ξ
∆Ξ Ξ
'
ΞΞ
'
f Ξ
'
(8.196)
The details are left to the exercises.
The generating function provides a Schläfli integral representation
gt,zExp
z
2
t t
1
n
J
n
zt
n
J
n
z
1
2Π
Exp
z
2
t t
1
t
t
n1
(8.197)
where the contour encloses the origin in a positive sense. For a circular contour, one obtains
Bessel’s integral
t
Θ
J
n
z
1
2Π
2Π
0
Expz SinΘ nΘ Θ (8.198)
or, more simply,
J
n
z
1
Π
Π
0
Cos
z SinΘ nΘ
Θ (8.199)
where z may be complex but n is integral.
8.3 Bessel Functions 295
8.3.2 Hankel Functions
Hankel functions are obtained by generalizing Bessel’s integral representation to noninte-
ger order Ν. Consider
f
Ν
z
1
2Π
Exp
z
2
t t
1
t
t
Ν1
(8.200)
where, except for avoiding the origin, we leave the integration path arbitrary for the moment.
If we apply Bessel’s differential operator to both sides
z
2
f
''
Ν
zzf
'
Ν
zz
2
Ν
2
f
Ν
z
1
2Π
z
2
4
t t
1
2
z
2
t t
1
z
2
Ν
2
Exp
z
2
t t
1
t
t
Ν1
(8.201)
the resulting integrand is a perfect differential, such that
z
2
f
''
Ν
zzf
'
Ν
z
z
2
Ν
2
f
Ν
z
1
2Π
$F
Ν
t,z (8.202)
where
F
Ν
t,zt
Ν
z
2
t t
1
Ν
Exp
z
2
t t
1
(8.203)
and where $ indicates the difference between values at the endpoints of integration. When
Ν!n is an integer, F
n
t,z is single-valued and the right-hand side vanishes for any closed
path; hence, f
Ν
is a solution to Bessel’s equation under those conditions. (Paths which do
not enclose the origin, though, degenerate to the trivial solution f
Ν
! 0.) When Ν is not
integral, we require a branch cut to interpret the integrated term. It is customary to cut
the t-plane below the negative real axis. We must also choose a contour for which F
Ν
t,z
has the same value at both ends for any z within a usefully large domain. The simplest
method is to choose semi-infinite contours that exploit the limiting properties
t ! 0
F
Ν
t,zGt
Ν
z
2t
Exp
z
2t
B 0forRez > 0 (8.204)
t !F
Ν
t,zGt
Ν
zt
2
Exp
zt
2
B 0forRez > 0 (8.205)
Thus, the two contours shown in Fig. 8.5 provide integral representations of the Hankel
functions
Rez > 0 H
Ν
z
1
Π
C
Exp
z
2
t t
1
t
t
Ν1
(8.206)
where the normalization is conventional. Similar representations for Imz < 0 can be
constructed by choosing a different cut for t and rotating the contours accordingly, as
detailed in the exercises. The Hankel functions satisfy the same recursion relations as the