Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
Подождите немного. Документ загружается.
316 8 Legendre and Bessel Functions
f
''
z
2
z
f
'
z
1
ΝΝ 1
z
2
f z0 f
Ν
zaj
Ν
zbn
Ν
z (8.363)
j
Ν
z
Π
2
z
1/ 2
J
Ν
1
2
z (8.364)
n
Ν
z
Π
2
z
1/ 2
N
Ν
1
2
z (8.365)
h
1
Ν
z j
Ν
zn
Ν
z
2
Π
z
1/ 2
H
1
Ν1/ 2
z (8.366)
h
2
Ν
z j
Ν
zn
Ν
z
2
Π
z
1/ 2
H
2
Ν1/ 2
z (8.367)
8.5.5.2 Recursion Relations
f
Ν1
z f
Ν1
z
2Ν1
z
f
Ν
z (8.368)
Ν f
Ν1
zΝ1 f
Ν1
z2Ν1 f
'
Ν
z (8.369)
z
z
Ν1
f
Ν
z
z
Ν1
f
Ν1
z (8.370)
z
z
Ν
f
Ν
z
z
Ν
f
Ν1
z (8.371)
8.5.5.3 Orthonormality
R
0
j
k
1
r j
k
2
rr
2
r R
2
k
2
j
k
1
R j
'
k
2
Rk
1
j
k
2
R j
'
k
1
R
k
2
1
k
2
2
(8.372)
8.5.5.4 Series
j
Ν
z
Π
2
z
2
Ν
m0
1
AΝ m
3
2
Am 1
z
2
4
m
(8.373)
8.5.5.5 Asymptotic Forms
Problems for Chapter 8 317
zj
Ν
z Sin
z
ΝΠ
2
(8.374)
zn
Ν
z Cos
z
ΝΠ
2
(8.375)
zh
1
Ν
z Exp
z
Ν 1Π
2
(8.376)
zh
2
Ν
z Exp
z
Ν 1Π
2
(8.377)
8.5.6 Fourier–Bessel Expansions
Exp
k ,
r4Π
0
j
krY
ˆ
k,Y
ˆr (8.378)
ExpkΞ SinΦ
m
J
m
kΞ
mΦ
(8.379)
Problems for Chapter 8
1. Rearrangement formulas for doubly infinite sums
a) Prove that the rearrangement formulas
m0
n0
a
n,m
p0
p
q0
a
q,pq
r0
"r/2#
s0
a
s,r2s
(8.380)
where
r even "r/2#
r
2
(8.381)
r odd "r/2#
r 1
2
(8.382)
is the floor function, are equivalent provided that the series is absolutely convergent.
b) Use this result to supply the missing steps in the derivation of the Legendre series.
2. Leibnitz’s formula
Prove Leibnitz’s formula
x
n
f xgx
n
m0
n
m
f
nm
xg
m
x (8.383)
for multiple derivatives of a product.
3. Special values for Legendre functions
a) Evaluate P
n
0.
318 8 Legendre and Bessel Functions
b) Evaluate P
'
n
1.
c) Use the Schläfli integral representation to evaluate P
Ν
1 for arbitrary Ν.
4. Legendre integrals
a) Use a recursion relation to evaluate
1
0
P
n
xx (8.384)
b) Evaluate
1
0
x
m
P
n
xx (8.385)
for positive integer m.
5. Legendre series
a) Evaluate
n0
x
n1
n 1
P
n
x (8.386)
b) Evaluate
1
1
P
n
x
1 x
2
x (8.387)
c) Produce Legendre expansions for ∆1 x.
6. Parseval relation for Legendre polynomials
Suppose that f x is expanded according to
f x
n0
a
n
P
n
x (8.388)
Express
1
1
f x
2
x in terms of the a
n
coefficients.
7. Legendre expansion of powers
Develop a Legendre expansion for powers, such that
x
m
m
n0
A
m,n
P
n
x (8.389)
by expressing the Legendre polynomials in terms of Rodrigues’ formula and using partial
integration to eliminate derivatives. You may find the beta function
Bp, q
1
0
t
p1
1 t
q1
t
ApAq
Ap q
(8.390)
useful, but a proof is not required.
Problems for Chapter 8 319
8. Legendre function P
Ν
z for arbitrary Ν
Show that Schläfli’s integral representation for the Legendre function of the first-kind sat-
isfies Legendre’s differential equation for arbitrary Ν provided that the contour and cuts are
defined properly.
9. Laplace’s integral representation for Legendre functions
a) Suppose that the contour used in Schläfli’s integral representation is a circle around z
with radius
z
2
1. Show that
P
n
z
1
Π
Π
0
z
z
2
1CosΦ
n
Φ (8.391)
b) Use Laplace’s integral representation to evaluate the Legendre generating function
gt,z
n0
t
n
P
n
z (8.392)
Thus, one obtains a generalization for complex z.
10. Generating function for associated Legendre functions
Derive the generating function for associated Legendre functions with m 2 0
g
m
t,x
m
2m!
2
m
m!
1 x
2
m/2
1 2tx t
2
m1/ 2
0
P
m,m
xt
(8.393)
from the generating function for Legendre polynomials. It is not shown often because it
appears cumbersome, but it can be useful in deriving relationships or performing integrals
involving associated Legendre functions.
11. An integral used in the normalization of associated Legendre functions
There are several methods for obtaining
1
1
1 x
2
x
Π
A 1
A
3
2
2
21
!
2
2 1!
(8.394)
for 2 0 (other than looking it up, of course). A relatively painless method that also
provides many related integral is based upon the beta function
Bp, q
ApAq
Ap q
(8.395)
where p, q are generally complex. For our purposes it will be sufficient to assume that p,
q are nonnegative real numbers so that we can employ the simplest integral representation
Ap
0
u
u
p1
u (8.396)
320 8 Legendre and Bessel Functions
Show that the substitution u ! x
2
and a transformation from Cartesian to polar coordinates
facilitates evaluation of the numerator as an integral over the first quadrant, such that
p, q 2 0 Bp, q4
Π/ 2
0
CosΘ
2p1
SinΘ
2q1
Θ (8.397)
It should now be a simple matter to obtain the desired integral using appropriate choices
of p, q and an obvious change of variable.
12. Recursion relations for associated Legendre functions
Many recursion relations for associated Legendre functions can be developed by differ-
entiating either the differential equation or a recursion relation for Legendre polynomials
m times. Use this technique to derive the following relations. For simplicity, assume that
m 2 0 for any P
,m
. Observe that a) varies the order, d) the degree, while b) and c) vary
both. It is probably easiest to derive these relations in the order listed.
a) P
,m2
xm 1
2x
1 x
2
P
,m1
x
1mm 1
P
,m
x0 (8.398)
b) xP
,m
m 11 x
2
1/ 2
P
,m1
P
1,m
0 (8.399)
c) P
1,m1
2 11 x
2
1/ 2
P
,m
P
1,m1
0 (8.400)
d) m 1P
1,m
2 1xP
,m
mP
1,m
0 (8.401)
13. m-raising and lowering operators for P
,m
a) Use the Rodrigues formula to derive the m-raising relation
P
,m1
1 x
2
1/ 2
P
'
,m
m
x
1 x
2
1/ 2
P
,m
(8.402)
b) Then use a recursion relation to deduce the corresponding m-lowering formula.
14. Special values for associated Legendre functions
Use the generating function to evaluate the following special values or limiting cases for
associated Legendre functions. Assume that m 2 0.
a) P
,m
1
b) P
,m
0
c) P
m,m
x for x CosΘ. Notice that this result can be used with the m-lowering operator
to generate the entire set of P
,m
x; this is a common algorithm.
15. Orthogonality of P
,m
with respect to m
The most useful orthogonality for P
,m
concerns differing but common m. Alternatively,
one can show that
1
1
P
,m
xP
,m
'
x
1 x
2
x
1
m
m!
m!
∆
m,m
'
,
m, m
'
> 0
(8.403)
expresses orthogonality between associated Legendre functions differing in order. This
result is generally less useful because orthogonality between azimuthal eigenfunctions
usually ensures matching order anyway.
Problems for Chapter 8 321
a) Show that the associated Legendre equation is a Sturm–Liouville equation with eigen-
value m and weight function wx1x
2
1/ 2
and, hence, provides this orthogonality
integral between eigenfunctions differing in order.
b) Obtain the normalization for P
,m
and use a recursion relation to step down m.
16. Recursion relations for Bessel functions
Suppose that f
Ν
zaJ
Ν
zbN
Ν
z where the coefficients a, b are independent of Ν or z.
Verify the following recursion relations.
a) f
Ν1
z
Ν
z
f
Ν
z f
'
Ν
z (8.404)
b)
z
z
Ν
f
Ν
z
z
Ν
f
Ν1
z (8.405)
c)
1
z
z
k
z
Ν
f
Ν
z
z
Νk
f
Νk
z (8.406)
d)
1
z
z
k
z
Ν
f
Ν
z
k
z
Νk
f
Νk
z (8.407)
17. Interleaving of roots for Bessel functions
Use the recursion relation
x
x
n
f
n
x
x
n
f
n1
x (8.408)
where f
n
xaJ
n
xbN
n
x is a linear combination of cylindrical Bessel functions with
constant coefficients to prove that there is one root of f
n1
between successive roots of f
n
and vice versa.
18. Rodrigues formula for Bessel functions
Derive a Rodrigues formula of the form
z
n
f
n
z
n
f
0
z (8.409)
where f
n
is a solution to Bessel’s equation (J
n
, N
n
, H
n
, etc.) and is a differential operator
applied n times to f
0
.
19. Orthonormality relations for Bessel functions and eigenfunction expansions
Piecewise continuous functions on a finite interval that satisfy linear homogeneous bound-
ary conditions can be expanded in terms of eigenfunctions for Bessel’s equation. Here we
develop the required orthonormality relations for simple boundary conditions and formu-
late the corresponding expansions.
a) Use the differential equations satisfied by J
Ν
k
1
Ξ and J
Ν
k
2
Ξ for arbitrary k
1
and k
2
to
show
Rk
2
J
Ν
k
1
RJ
'
Ν
k
2
Rk
1
J
Ν
k
2
RJ
'
Ν
k
1
R k
2
1
k
2
2
R
0
J
Ν
k
1
ΞJ
Ν
k
2
ΞΞ Ξ
(8.410)
for Ν20. Differentiation of this result and application of recursion relations or the
differential equations can then help with specific boundary conditions.
322 8 Legendre and Bessel Functions
b) Derive the orthonormality relation
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
(8.411)
where J
Ν
k
Ν,n
R0 defines the eigenvalues k
Ν,n
for Dirichlet boundary conditions.
Provide an explicit expression for the Bessel expansion of an arbitrary function satis-
fying Dirichlet boundary conditions.
c) Derive the orthonormality relation
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
(8.412)
where J
'
Ν
k
Ν,n
R0 defines the eigenvalues k
Ν,n
for Neumann boundary conditions.
Provide an explicit expression for the Bessel expansion of an arbitrary function satis-
fying Neumann boundary conditions assuming that Ν0.
d) How is the Bessel expansion using Ν0 for an arbitrary function satisfying Neu-
mann boundary conditions affected by the null eigenvalue k
0
0? Show that the
corresponding eigenfunction is orthogonal to those with k
n
> 0 and formulate the
appropriate expansion explicitly.
20. Continuum orthonormality for Bessel functions
The continuum orthonormality relation
0
J
Ν
kΞJ
Ν
k
'
ΞΞ Ξ
∆k k
'
kk
'
(8.413)
for Bessel functions with Ν20 on a semi-infinite range can be derived by careful analysis
of the limit of the result for a finite interval
R
0
J
Ν
k
1
ΞJ
Ν
k
2
ΞΞ Ξ R
k
2
J
Ν
k
1
RJ
'
Ν
k
2
Rk
1
J
Ν
k
2
RJ
'
Ν
k
1
R
k
2
1
k
2
2
(8.414)
as R !. This result should have been obtained in the preceding problem. Use a recursion
relation to eliminate the derivatives first and then substitute the leading asymptotic behav-
ior for the remaining Bessel functions. (Why eliminate derivatives first?) After some alge-
braic manipulation, more easily performed with than by hand, one should
recognize a familiar nascent delta function.
21. Parseval relation for Bessel functions
Suppose that f x is expanded according to
f x
n1
a
n
J
Ν
k
Ν,n
x ,J
Ν
k
Ν,n
R0 (8.415)
in the range 0 x R. Express
R
0
f x
2
x x in terms of the a
n
coefficients.
Problems for Chapter 8 323
22. Summation formula for Bessel functions
Use the generating function to evaluate J
n
x y for integer n.
23. Laplace transform of Bessel functions
Use Bessel’s integral for J
n
x to evaluate the Laplace transform for Bessel functions of
integral order.
24. Bessel’s integral for noninteger Ν
Show that
J
Ν
z
1
Π
Π
0
Cos
z SinΘ ΝΘ
Θ
SinΝΠ
Π
1
Exp
z SinhsΝs
s (8.416)
for a suitable domain of z.
25. Plane wave in cylindrical Bessel functions
Show that a plane wave can be expanded in terms of cylindrical Bessel functions according
to
Exp
kΞ SinΦ
m
J
m
kΞ
mΦ
(8.417)
26. Leading asymptotic behavior of Hankel functions
Apply the method of steepest descent to deduce the leading asymptotic behavior of Hankel
functions from their integral representations. Then deduce the corresponding behavior of
Bessel and Neumann functions. For simplicity assume that zDΝ(why?).
27. Power series for Neumann functions
Complete the derivation of the power series for Neumann functions with integer n 2 0
using the fact that gamma functions have simple poles with known residues at negative
integers.
28. Wronskians for Bessel functions
Recall that the Wronskian
Wf,g f zg
'
zgz f
'
z-
1
pz
(8.418)
for two independent solutions f , g to a Sturm–Liouville equation
Λwz f z0 (8.419)
Λwzgz0 (8.420)
generalized to complex z where
z
pz
z
qz (8.421)
takes the form
Wf,g-
1
pz
(8.422)
Confirm the validity of this result for Bessel functions and identify pz. Then evaluate
the following Wronskian relations between various Bessel functions. Also, express these
324 8 Legendre and Bessel Functions
Wronskians for parts a) and b) as combinations of products of Bessel function of different
orders, without explicit derivatives.
a) WJ
Ν
,J
Ν
. Explain the result for Ν!n, where n is an integer.
b) WJ
Ν
,N
Ν
c) WH
1
Ν
,H
2
Ν
.
29. Recursion relations for spherical Bessel functions
Suppose that f
Ν
zaj
Ν
zbn
Ν
z where the coefficients a and b are independent of
Ν or z.
a) Starting with the corresponding relations for cylindrical Bessel functions, verify the
basic recursion relations:
f
Ν1
z f
Ν1
z
2Ν1
z
f
Ν
z (8.423)
Ν f
Ν1
zΝ1 f
Ν1
z2Ν1 f
'
Ν
z (8.424)
f
Ν1
z
Ν
z
f
Ν
z f
'
Ν
z (8.425)
f
Ν1
z
Ν1
z
f
Ν
z f
'
Ν
z (8.426)
Then demonstrate the following variations:
b)
z
z
Ν1
f
Ν
z z
Ν1
f
Ν1
z (8.427)
c)
z
z
Ν
f
Ν
z z
Ν
f
Ν1
z (8.428)
d)
1
z
z
k
z
Ν1
f
Ν
z z
Νk
f
Νk
z (8.429)
e)
1
z
z
k
z
Ν
f
Ν
z
k
z
Νk
f
Νk
z (8.430)
30. Poisson integral representation for spherical Bessel functions
Use
j
kr
2
1
1
ExpkrxP
xx (8.431)
derived in the text to obtain the Poisson integral representation
j
Ρ
Ρ
2
1
!
Π
0
Cos
Ρ CosΘ
SinΘ
21
Θ (8.432)
for real Ρ and nonnegative integer .
Problems for Chapter 8 325
31. Orthonormality relations for spherical Bessel functions
a) Use the differential equations satisfied by j
k
1
r and j
k
2
r for arbitrary k
1
and k
2
to
show
R
2
k
2
j
k
1
R j
'
k
2
Rk
1
j
k
2
R j
'
k
1
R k
2
1
k
2
2
R
0
j
k
1
r j
k
2
rr
2
r
(8.433)
for 2 0.
b) Derive the orthonormality relation
j
k
,n
R
0
R
0
j
k
,m
r
j
k
,n
r
r
2
r
R
3
2
j
1
k
,n
R
2
∆
m,n
(8.434)
for Dirichlet boundary conditions. Provide an explicit expression for the spherical
Bessel expansion of an arbitrary function satisfying Dirichlet boundary conditions.
c) Derive the continuum orthonormality relation
2
Π
0
j
kr j
k
'
rr
2
r
∆k k
'
kk
'
(8.435)