Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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14.5
CONFLUENT
HYPERGEOMETRIC
FUNCTIONS
423
His observation
of
the variation
of
the proper
motion
of
the stars Sirius and Procyon
led
him to posit the existence
of
nearby, large, low-luminosity stars called dark companions.
Between1821and1833he cataloguedthe positionsof about75,000stars,publishing his
measurements in detail. One
of
his
most
important contributions to astronomy was the
determinationof the distanceto a starusingparallax.This methoduses triangulation,or the
determinationof theapparentpositions of a distant objectviewedfrom twopointsa known
distanceapart,in thiscasetwodiametrically opposedpointsoftheEarth's orbit.The angle
subtended by the baseline
of
Earth's orbit, viewed from the star's perspective, is known
as the star's parallax. Before Bessel's measurement, stars were assumed to be so distant
that their parallaxeswere too smallto measure,and it was further assumedthatbright stars
(thoughtto be nearer) would have the largest parallax. Besselcorrectly reasoned that stars
with large proper motions were
more
likely to be nearby ones and selected such a star, 61
Cygni, for his historic measurement. His measured parallaxfor that stardiffers by less than
8% from the currently acceptedvalue.
Given such an impressive record in astronomy, it seems only fitting that the famous
functions that bearBessel's
name
grew out
of
his investigations
of
perturbationsin planetary
systems. He showed that such perturbations could be divided into two effects and treated
separately: the obvious direct attraction due to the perturbing planet and an indirect effect
caused by the
sun's
response to theperturber's force.
The
so-called Bessel functions then
appear as coefficients
in the series treatment
of
the indirectperturbation. Although special
cases
of
Bessel functions were discovered by Bernoulli,Euler, and Lagrange, the systematic
treatment by Bessel clearly established his preeminence, a fitting tribute to the creator
of
the
most
famous functions in mathematicalphysics.
14.5.1 BesselFunctions
Bessel
differential
The Bessel
differential
equation
is usually written as
equalion
1/
I,
( V
2)
W +
-w
+
1-
- w = 0
Z Z2
(14.41)
As in the example above, the substitution
w =
zI'e-~'
f(z)
transforms (14.41)
inlo
d
2
f
(2/k
+I )
df
[/k
2
- v
2
~(2/k
+I) 2
IJ
f 0
-+---2~-+
-
+~+
=,
di
z ~
~
z
which, if we set /k =v and
~
=i, reduces to
r +
CV:
I _
2i)
f'
_
(2V;
I);
f =
O.
Making the further substitution 2;z = t, and multiplying
out
by t, we obtain
d
2f
df
t-
2
+(2v+I-t)--(v+!)f=O,
dt
dt
424 14. COMPLEX ANALYSIS OFSOLOES
which is in the
fonn
of
(14.37) with a = v +1and y =2v +1.
Thns, solutions
of
the Bessel equation, Equation (14.41), can be wtitten as
constantmultiples
of
zVe-iz<I>(v
+
1,
2v +I;
2iz).
With propernonnalization,we
Bessel
function
ofthe define the Bessel
function
of
the
first
kind
of
order
v as
first
kind
1
(Z)v.
1
Jv(z) = - e-"<I>(v +
2'
2v + 1;
2iz).
f(v
+1) 2
Using Equation (14.38) and the expansion for
e-
iz,
we can show that
(
Z)V
00
(_I)k
(Z)2k
Jv(z)=
2"
~k!f(V+k+l)
2"
.
(14.42)
(14.43)
The
second linearly independent solution can be obtained as usual and is propor-
tional to
ZI-(2v+1)
G)
v
e-iz<I>(v
+1-(2v +I) +I, 2
.,
(2v +
I);
2iz)
=C
G)
-v
e-iz<I>(_v +
1,
-2v
+1;
2iz)
=
C1-
v(z)
,
provided that
1-
y =
1-
(2v +1) =
-2v
is not an integer. When v is an integer,
1-
n(z)
=
(_I)n
In(z) (see Problem 14.25). Thus,when v is anoninteger, the most
general solution is
of
the form
AJv(z)
+
B1-
v(z).
How do we find a second linearly independent solution when v is an integer
n? We first define
Bessel
function
ofthe
second
kind,
or
Neumann
function
J»
(z) cos
V1f
-
1-
v(z)
Yv(z)
= . ,
sm
uzr
(14.44)
called the Bessel function
of
the second kind, or the
Nenmann
function. For
noninteger
v this is simply a linear combinatiou
of
the two linearly independent
solutious.
For
integer v the functionis indeterminate. Therefore, we use I'Hopital's
rule and define
. 1 .
[aJ
v
na1-V)
Yn(z)
==
Ion Yv(z) = - Ion - -
(-1)
--
.
v-+n
1{
v--+-n
8v
8v
Equation (14.43) yields
aJ
v
= Jv(z) In
(!:)
_
(!:)V
f(
_1)k
W(v +k +1)
(!:)2k,
av 2 2
k=O
k!f(v
+k +1) 2
where W(z) =
(dldz)
In
I'(z),
Sintilarly,
a1-
v
=
-1-
v(z)
In
(!:)
+
(!:)-v
f
W(-v
+k +1)
(!:)2k.
av 2 2
k~O
k!f(-v
+k+
1) 2
14.5
CONFLUENT
HYPERGEOMETRIC
FUNCTIONS
425
Substituting these expressions in the definition of Y
n
(z) and using L
n
(z) =
(-I)nJn(z),
we obtain
The natural log term is indicative
of
the solution suggested by Theorem 14.2.6.
Since
Yv(z) is linearly independent of Jv(z) for any v, integer or noninteger, it is
convenienttoconsider{Jv(z), Yv(z)} as abasis
of
solutions for the Bessel equation.
Another basis of solutions is defined as
Bessel
function
of
the
third
kind
or
Hankel
function
2
(Z)
I
(z)n~
k
qr(n+k+l)
(Z)2k
Yn(z)=;Jn(z)ln
2:
-;
2:
6(-1)
k!I'(n+k+l)
2:
_~(~)-n
-I
n~
-I
k
qr(k-n+l)
er
T( 2
()
6(
)k!I'(k - n +I) 2
(14.45)
(14.46)
which are called Bessel functions of the third kind, or
Hankelfunctions.
Replacing z by
iz in the Bessel equation yields
whose basis
of
solutions consists of multiples of Jv(iz) and
Lv(iz).
Thus, the
modified Bessel fnnctions
of
the
first
kind
are defined as
.
(Z)v
00
I
(Z)2k
I -
-mvj2],
. -
v(z) = e u(zz) -
2:
t;k!I'(v +k +I)
2:
.
Similarly, the modified Bessel fnnctions
of
the
second kindare defined as
T(
Kv(z) =
2'
[I-v(z)
- Iv(z)].
SIll
v:rr
When v is
anintegern,
In = L
n,
and K
n
is indeterminate. Thus, we define Kn(z)
as
limv-->n
Kv(z). This gives
Kn(z) =
(_I)n
lim
[aL
v
_
aI
v].
2 v--+n
Bv Bv
which has the power-series representation
426 14.
COMPLEX
ANALYSIS
OF
SOLOES
We can obtain a recorrence relation for solntions
of
the Bessel eqnation as
follows.
If
Zv(z) is a solntion
of
order v,
then
(see Problem 14.28)
recurrence
relation
for
solutions
of
the
BesselOE
and
v d v
Zv-l
=
C2C
-[z
Zv(z)].
dz
If
the constants are chosen in snch a way that Zv,
Z-v,
Zv+l,
and
Zv-l
satisfy
their appropriate series expansions, then Cl
=
-I
and C2 = 1. Carrying ont the
differentiation in the equations for
Zv+l
and
Zv-l,
we obtain
v dZ;
Zv+l
=
-Zv
-
-d
'
z z
v sz;
Zv-l
=
-Zv
+
--.
z dz
(14.47)
Adding these two eqnations yields the recursionrelation
2v
Zv-l(Z)
+Zv+l(Z) =
-Zv(z),
z
where Zv(z) can be any
of
the three kinds
of
Bessel functions.
14.6 Problems
(14.48)
for
n 2:
O.
14.1. Show that the solution
of
w' +w/Z2 = 0 has an essential singularity at
z
=0.
14.2. Derive the recursion relation
of
Equation (14.7) and express it in terms
of
the indicial polynomial, as in Equation (14.9).
14.3. Find the characteristic exponent associated with the solution
of
ui"
+
p(z)w
'
+
q(z)w
= 0 at an ordinary point [a point at which
p(z)
and q(z) have no
poles]. How many solutions can you find?
14.4.
The
Laplace equation in electrostatics when separated in spherical coordi-
nates yields a DE in the radial coordinate given by
~
(x
2dY
) _
n(n +
I)y
= 0
dx dx
Starting with an infinite series
of
the form (14.6), show that the two independent
solutions
of
this
ODE
are
of
the form x
n
and
x-
n
-
1
.
14.5. Findthe indicialpolynomial,characteristicexponents,and recursionrelation
at both
of
the regular singular points
of
the Legendre equation,
2z a
wI! -
_-w'
+
--w
=
O.
I-
Z
2
l-
z
2
What is ai, the coefficient
of
the Laurent expansion, for the point z = +I?
F(
-a,
fJ; fJ;
-z)
=
(I
+
z)",
14.6
PROBLEMS
427
14.6. Show thatthe substitution z =
lit
transforms Equation (14.13) into Equa-
tion (14.14).
14.7. Obtain the indicial polynomial
of
Equation (14.14) for expansion about
t =
O.
14.8. Show thatRiemann DE represents the most general second order Fuchsian
DE.
14.9. Derive the indicial equation for the Riemann DE.
14.10. Show that the transformation v(z) =
ZA(Z
-
I)"w(z)
changes the pairs
of
characteristic exponents 0.1, A2), (iLl, iL2), and
(VI,
V2) for the Riemann DE to
(AI +A,
A2
+A),
(J.L1
+u; iL2 +iL), and (VI - A-
u,
V2
- A- iL).
14.11. Go through the steps leading to Equations (14.24), (14.25), and (14.26).
14.12. Show that the elliptic function
of
the first kind, defined as
/0
" /2
de
K(z)
= ,
o
JI-
Z
2
s
in
2
e
can be expressed as (n
12)F(~,
~;
I; Z2).
14.13. By differentiating the hypergeometric series, show that
d
n
r(a
+
n)r(fJ
+
n)r(y)
dz
n
F(a,
fJ;
y;
z) =
r(a)r(fJ)r(y
+n)
F(a
+n, fJ+n; y +n; z).
14.14. Use direct substitution in the hypergeometric series tu show that
1 1.
3.
2
1'_1
F(2'
2'
2'Z)
=
-sm
z,
z
I
F(I,
I; 2;
-z)
=
-In(1
+z).
z
14.15. Show that the substitution v(z) =
ZT
w(l/z)
[see Equation (14.28)] trans-
forms the
HGDE
into Equation (14.29).
14.16. Consider the function v(z)
'"
zT(I -
z)'
F(al,
fJI; Yl;
liz)
and assume
that it is a solution of HGDE. Find a relation among
r, s,
aI,
fJt, and
Yl
suchthat
v(z) is written in terms
of
three parameters rather than five. In particular, show
that one possibility is
v(z) =
za-
y
(I -
z)y-a-P
F(y
- a, I - a; I +fJ - a;
liz).
Find all such possibilities.
428
14.
COMPLEX
ANALYSIS
OF
SOLOES
14.17. Show that the Jacobifunctions are related to the hypergeometric fooctions.
14.18. Derive the expression for the Jacobi function
of
the second kind as given
in Equation
(14.33).
14.19. Show that z =
00
is not a regular singular point
of
the CHGDE.
14.20. Derive the confluent hypergeometric series from hypergeometric series.
14.21. Show thatthe
Weber-Hermite
equation,
u"
+(v+
~
-
!Z2)u
= ocan be
transformedinto the CHGDE. Hint: Make the substitution
u(z)
= exp(-!Z2)v(Z).
14.22. The linear combination
. _
I'(l
-
y)
.
I'(y
- I)
l-y
_ .
ljI(a,
y,
z) = I)
<!>(a,
y, z) +
I'(a)
Z
<!>(a
y +
1,2
- y, z)
I'(a
- y +
j,
a
is also a solntion
of
the CHGDE.Show thatthe Hermitepolynomialscan be written
as
(
Z) "
nlz
2
H" .,fi = 2
1jI(-2'
2;
2)'
14.23. Verify that the error fooction erf'(z) =
J~
e-t'dt
satisfies the relation
erf'(z) =
z<!>(~,
~;
_Z2).
14.24. Derive the series expansion
of
the Bessel function
of
the first kind from
that
of
the confluenthypergeometric series and the expansion
of
the exponential.
Check your answer by obtaining the same result by snbstituting the power series
directly in the Bessel DE.
14.25. Show that 1-,,(z) =
(-I)"J,,(z).
Hint:
Let
v =
-n
in the expansion
of
J
v
(z) and use I'Cm) =
00
for a nonpositive integer
m,
14.26.
In
a potential-free region, the radial part
of
the Schrodinger equation re-
duces to
d
2
R +
~
dR
+
[I..
_
:!...]
R =
O.
dr
2
r dr r
2
Write the solutions
of
this DE in terms
of
Bessel functions. Hint: Substitute R =
spherical
Bessel
uj.p.
These solutions are called
spherical
Bessel functions.
functions
14.27. Theorem 14.2.6 states that ooder certain conditions, linearly independent
solutions
of
SOLDE at regular singolar points exist even though the difference
between the characteristic exponents is an integer.
An example is the case
of
Bessel fooctions
of
half-odd-integer orders. Evaluate the Wronskian
of
the two
linearly independentsolutions,
J
v
and
J-
v
,
of
the Besselequationand show that it
vanishes
only
ifv
is an integer.This shows, in particular, that
J,,+1/2
and 1-,,-1/2
are linearly independent. Hint: Consider the value
of
the Wronskian at z = 0, and
use the formula
I'(v)I'(l
- v) =
x]
sin vrr.
14.6
PROBLEMS
429
14.28. Show that
z±V(djdz)[z'FvZv(z)]
is a solution of the Bessel equation of
order v
± I
if
Z,
is a solution of order v.
14.29. Use the recursion relation
of
Equation (14.47) to prove that
(
I
d)m
u v m
~
dz
lz
Zv(z)]
= z -
Zv-m(Z),
I d m
(~dJ
[z-VZv(z)]
=
(-I)mz-V-mZv+m(z).
14.30. Using the series expansion of the Bessel function, write JI/2(Z) and
Ll/2(Z)
in terms of elementary functions. Hint: First show that
r(k
+
~)
=
v'IT
(2k +
1)!j(k!2
2
k+l).
14.31. From the results of the previous two problems, derive the relations
L
n
- I/2(Z) =
[fr
l
/
2
(~:y
("o;Z)
,
I
n
+I/2(Z) =
[fzn+I/2
(_~:y
("i~Z)
.
14.32. Obtain the following integral identities:
(a) fzV+l
Jv(z)dz
= zV+lJv+l(Z).
(b) f
z-v+IJv(z)dz
=
-z-v+lJv_l(Z).
(e) f
ZIL+l
Jv(z)dz
= ZIL+
1
Jv+l(Z) +(I-' -
v)ZIL
Jv(z)
-
(1-'2
- v
2)
f
ZIL-
1
Jv(z)
dz;
and evaluate
(d)
fZ3Jo(z) dz:
Hint: For (c) write ZIL+
1
= ZIL-
V
z
v+
1
and use integration by parts.
14.33. Use Theorem 14.2.6and the fact that
In(z)
is entire to show that for integer
n, a second solution to the Bessel equation exists and can be written as Y
n
(z) =
I
n
(z)[/n
(z) +
en
ln
z], where
In(z)
is analytic about z =
O.
14.34. (a) Show that the Wronskian
W(Jv,
Z; z)
of
J
v
and any other solution Z
of the Bessel equation, satisfies the eqnation
d
dz[zW(Jv,
Z; z)] =
o.
430 14.
COMPLEX
ANALYSIS
OF
SOLOES
(b) For some constant A, show that
~
[~]
_ W(z) _ _
A_
dz
J
v
-
J;(z)
-
zJ;(z)'
(c) Show that the general second solution
of
the Bessel equation can be written as
14.35. Spherical Bessel functions are defined by
Let
il(Z)
denote a spherical Bessel function
"of
some
kind."
By direct differenti-
ationandsubstitution in theBessel equation, showthat
(c) Combine the results of parts (a) and (b) to derive the recursion relations
2/
+1
il-t(Z)
+
fi+l(z)
=
--fj(z),
z
df;
/fi-I
(z) - (/ +l)fi+1(z) = (2/ +1)
dz
.
14.36. Show that W(Jv, Y
v
;
z) = 2/(:n:z), W(HJ!), HS
2
);
z) = 4/(i:n:z). Hint:
Use Problem 14.34.
14.37. Verify the following relations:
(a) Y
n+I/2(Z)
=(_1)n+1 L
n-I/2(Z),
L
n-I/2(Z)
=
(-1)"
I
n
+I/2(Z).
J
v
(z) - cos
v:n:
Lv
(z)
(b)
Lv(z)
= sin
v:n:
Jv(z) +cos v:n:Yv(z) = . .
sm vn
(c)
Y-n(z)
=
(-l)nyn(z)
in the limit v --> n in part (b).
14.38. Use the recurrence relation for the Bessel function to show that JI (z) =
-Jij(z).
14.39. Let u = J
v
().z) and v = J
v
(jLz). Multiply the Bessel DE for u by v[z and
that
of
v by
uf
z, Subtract the two equations to obtain
0.
2
_ jL2)zuv =
~
[z
(u
dv
_ v
dU)]
.
dz dz dz
(a) Write the above equation in terms
of
J
v
(}"z) and J
v
(jLz) and integrate both
sides withrespectto Z.
ois w =
14.6
PROBLEMS
431
(b) Now divide both sides by A
2
-
,,2
and take the limit
as"
-->
A.
Youwill need
to use L'Hopital's rule.
(c) Substitute for
J;'(AZ) from the Bessel DE and simplify to get
f
z[Jv(Az)fdz
=
Z;
{[j;(AZ)]2
+(
1
-
A~:2)
[Jv(Az)f}
.
(d)Finally, let X=
xvn/a,
where
r.;
is the nth root of J
v,
and use Equation (14.47)
toarriveat
{a 2
(X
vn
) a
2
2
io
zJ
v
---;;z
dz =
2Jv+I
(x
vn).
14.40. The generatingfunction for Bessel functions of integer order is
exp[~z(t
I/t)].
To see this, rewtite the generating function as
e"/2
e-
z/21,
expand both
factors, and wtite the product as powers of t", Now show that the coefficient
of
t" is simply
In(z).
Finally, use
Ln(z)
=
(-I)nJn(z)
to derive the formula
exp[~z(t
-
I/t)]
=
L~-oo
JIl(z)t
n.
14.41. Make the substitutions z = fJt
Y
and w =
tau
to transform the Bessel DE
into t
2
d
2
u
+ (2", +
l)t
du +(fJ2
y
2
t2
Y
+
",2
_ v
2
y
2)u =
O.
Now show that
dt
2
dt
Airy's differential
equation
ii - tu = 0 has solutions of the form
hf3(~it3/2)
and
LI/3(~it3/2).
d
2
w
14.42. Show that the general solution
of
dt
2
+
t[AJv(e
l/')
+
BYv(e
l/')].
14.43. Transform
dw/dz
+ w
2
+ zm = 0 by making the substitution w =
(d/dz)
In v. Now make the forther substitutions
v=
u..;z
and
t =
_2_
z
!+O/2)m
m+2
to show that the new DE can be transformed into a Bessel equation of order
I/(m
+2).
14.44. Starting with the relation
exp[~x(t
-
l/t)]
exp[~Y(t
-
l/t)]
=
exp[~(x
+
y)(t
-
l/t)]
and the factthat the exponentialfunction is the generatingfunction for I
n
(z), prove
the "additiontheorem" for Bessel functions:
00
JIl(x +y) = L
h(X)Jn-k(Y)·
k=-oo
432 14.
COMPLEX
ANALYSIS
OF
SOLDES
Additional Reading
1. Birkhoff,G. andRota,G.-C. OrdinaryDifferentialEquations, 3rded.,Wiley,
1978.
The
first two sections
of
this chapterclosely follow theirpresentation.
2. Dennery,P.and Krzywicki,
A.Mathematicsfor Physicists, Harperand Row,
1967.
3. Watson, G.
A Treatise on the Theory
of
Bessel Functions,
2nd
ed., Cam-
bridge University Press, 1952. As the namesuggests, the definitive text and
reference on Bessel functions.