Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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7.1
GENERAL
PROPERTIES
173
Then F
n
(x) is a polynomial
of
degree n in x and is orthogonal-on the inter-
val (a, b), with weight w(x)---to any polynomial Pk(X)
of
degree k < n, i.e.,
J:
Pk(x)Fn(x)w(x)
dx
= Ofor k < n. These polynomials are collectively called
classical onhogonalpolynomials.
Before proving the theorem, we need two
lemmasf
7.1.2.
Lemma.
Thefollowing identity holds:
Proof
See Problem 7.1.
m:S
n.
D
7.1.3.
Lemma.
All
the derivatives d/"
jdxm(ws
n)
vanish at x
= a and x =
b,for
all values
of
m < n.
Proof
Set k = 0 in the identity of the previous lemmaand let P,o = 1. Then we
d
m
have
--(ws
n)
= ws
n-
m
Fsm- The RHS vanishes
atx
= a and x = b due to the
dx'" -
third condition stated in the theorem. D
Proof
of
the theorem. .We prove the orthogonality first. The proofinvolves multi-
ple use of integration by parts:
l
b
lb
I d
n
pk(x)F"(x)w(x)dx
=
Pk(x)-
[-n
(ws
n)]
wdx
a a W dx
l
b
d
[d
n
-
1
]
=
Pk(X)-
---I
(uis")
dx
a
dx
dx»:
d
n-
I
I
b
l
b
d
d"-l
= Pk(X)
--I
(ws") - d
Pk
-d
1(ws
n)
dx.
dxv:
a a X x
ll
-
~
=0 by Lemma7.1.3
This shows that each integration by parts transfers one differentiation from uis" to
Pk and introduces a minus sign. Thus, after k integrations by parts, we get
l
b
lb
d
k
Pk d
n
-
k
a
Pk(x)Fn(x)w(x)dx
=
(_I)k
a dxk
dxn_k(wsn)dx
l
b
d
[d
n-
k-
I
]
d"-k-l
I
b
= C a dx
dx"
k 1 (ws
n)
dx = C
dx"
k 1 (ws
n)
a = 0,
2Recall that P5k is a generic polynomial with degree less than or equal to k.
174
7.
CLASSICAL
ORTHOGONAL
POLYNOMIALS
where we have used the fact that the kth derivative of a potynomial of degree k
is a constant. Note that n - k - 1
~
0 because k < n, so that the last line of the
equation is well-defined.
Toprove the firstpartof the theorem, we use Lemma7.1.2 with
k = 0 and m =
d
n
1 d
n
n to get
-(ws
n)
= wP<n,orFn(x) =
--(ws
n)
= Psn-Toprove that Fn(x)
dxn
-
ui
dx"
-
is a polynomial
of
degree preciselyequal
ton,
we write Fn(x) =
P~n-'
(x)+knx
n,
multiply both sides by w(x)Fn(x), and iotegrate over (a, b):
l
b
[Fn(x)]2
w(x)dx
= l
b
P~n_,Fn(x)w(X)
dx +k
n
l
b
x
n
Fn(x)w(x)dx.
The LHS is a positive quantity because both
w(x)
and [Fn(x)]2 are positive, and
the first iotegral on the RHS vaoishes by the first part of the proof. Therefore, the
second term on the RHS cannot be zero.
In particular, k
n
t=
0, and F
n
(x)
is
of
degreen. D
It
is customaryto iotroduce a normalizationconstantio the definition of F
n
(x),
and write
1 d
n
Fn(x) =
--
__
(ws
n).
Knwdx
n
(7.2)
generalized
Rodriguez
formula
differential
equation
for
classical
orthogonal
polynomials
This equationis calledthe generalized
Rodriguez
fonnula.
For historicalreasons,
different polynomial functions are normalized differently, which is why
K.
is
introduced here.
From Theorem 7.1.1 it is clear that the sequence
Fo(x), F, (x), F2(X), . " of
polynomialsforms an orthogonalset of polynomials on
[a, b] with weight function
w(x).
All the varieties
of
classical orthogonal polynomials were discovered as solu-
tions of differential equations. Here, we give a siogle generic differential equation
satisfied by all the
Fn's. The
proof
is outlioed in Problem7.4.
7.1.4. Proposition. Let k, be the coefficientofx in Fl (x) and (72 the coefficient
of
x
2
in sex). Then the orthogonalpolynomials F
n
satisfy the differential equation
3
We shall study the differential equation above in the context of the Sturm-
Liouville problem (see Chapters 18 and 19), which is an eigenvalue problem in-
volving differential operators.
3A
prime
is a
symbol
for
derivative
with
respect
tox.
7.2
CLASSIFICATION
175
7.2 Classification
Let us now investigate the consequences of vatious choices of s
(x).
We start with
Ft (x), and note that it satisfies Equation (7.2) withn = 1:
1 d
Ft(x)
=
--(ws),
Kiui
dx
or
(7.3)
which can be integrated to yield
ws =
Aexp(f
KtFt(x)dx/s)
where A is a
constant. On the other hand, being a polynomial of degree 1,
Fl
(x)
can be written
as
F;(x) = klX +k;.
It
follows that
(I
Kl (klX +
kJ)
)
w(x)s(x)
=
Aexp
s
dx,
w(a)s(a) = 0 = w(b)s(b).
Nextwe lookat the three choices for s(x): a constant, a polynomial of degree
1, and a polynomial of degree
2.
For
a constant s
(x),
Equation (7.3)
can
be easily
integrated:
(I
Kl(klX
+
k'))
(I
)
w(x)s(x)
=
Aexp
s 1 dx
=Aexp
(2ax+fJ)dx
= Aeax2+px+c = Beax2+px.
The interval (a, b) is determined by w(a)s(a) = 0 = w(b)s(b), which yields
Beua2+pa
= 0 =
Beub2+Pb.
The only way that this equality can hold is for a and
b to be infinite. Since a < b, we musttake a =-00 and b =
+00,
in which case
a <
O.
With y = .JIiiT(x+fJ/(2a)) and choosing B =s exp(fJ2
/(4a)),
we obtain
w(y)
= exp(
_y2).
We also take the constant
sto
be 1. This is always possible by
a proper choice of constants such as
B.
lfthe
degree
of
sis
1, then
s(x)
=
O'IX
+0'0 and
where
y = Klkl/O'I, P = Klk1JO'I - KlklO'O/O'[, and B is A modified by
the constant
of
integration. The last equation above must satisfy the boundary
conditions at
a and b: B(O'la +
O'o)Peya
= 0 = B(O'lb +
O'O)Peyb,
which give
a =
-0'0/0'1,
P > 0, Y < 0, and b =
+00.
With appropriate redefinition of
vatiables and parameters, we
can
write
w(y)
= yVe-Y, v >
-1,
and s(x) =x,
a =0, b =
+00.
176
7.
CLASSICAL
ORTHOGONAL
POLYNOMIALS
fL
v
w(x)
Polynomial
0
0
I
Legendre, Pn(x)
J.-!
J.-!
(I - x
2
jA
- l/2
Gegenbauer,
C~
(x), J. > -!
1 1
(I
- x
2
)- 1/2
Chebyshev
of
the first kind, Tn(x)
-2 -2
1 1
(I - x
2
)1/2
Chebyshev of the second kind, Un(x)
2 2
Table7.1
Special
casesof
Jacobi
polynomials
Similarly, we can obtain the weight function and the interval of integrationfor
the case when
s
(x)
is of degree 2. This result, aswell asthe results obtainedabove,
are collected in the following proposition.
7.2.1. Proposition.
Ifthe conditions
of
Theorem 7.1.1 prevail, then
(a) For
s(x)
of
degree zero we get
w(x)
=
e-
x'
with
s(x)
= I, a =
-00,
and
b =
+00.
The resulting polynomials are called Hermite polynomials
and
are denoted by H
n
(x).
(b) For
s(x)
of
degree 1,we obtain
w(x)
=
xVe-
x
with v
>
-1,
s(x)
= x,
a
= 0, and b =
+00.
The resulting polynomials are called Laguerre
polynomials and are denoted by
L~(x).
(c)
Fors(x)
of
degree 2, we get
w(x)
= (1
+x)I'(I-x)V
with u.,» >
-1,
s(x)
= 1- x
2
,
a =
-1,
and
b =+1. The resulting polynomials are called
Jacobipolynomials
and
are denoted by p/:. v (x).
Jacobipolynomials are themselves divided into other subcategories depending
on the values of
fLand v. The mostcommon and widelyused of these are collected
inTable 7.1. Note that the definition
of
each
of
the precedingpolynomialsinvolves a
"standardization," which boils down to a particular choice
of
K
n
in the generalized
Rodriguez formula.
7.3 Recurrence Relations
Besides the recurrence relations obtained in Section 5.2, we
can
use the differen-
tial equation of Proposition 7.1.4 to construct new recurrence relations involving
derivatives. These relations apply only to classical orthogonal polynomials, and
notto
general ones. We start with Equation (5.12)
Fn+l(X)
= (anx +fJn)Fn(x) +)'nFn-l(X), (7.4)
(7.5)
7.3
RECURRENCE
RELATIONS
177
differentiate both sides twice,
and
substitnte for
the
secoud derivative from the
differeutial equation
of
Propositiou 7.1.4. This will yield
2wsanF~
+
[an:X
(ws)
+WAn(anX +fin)]
t;
- WAn+!
Fn+1
+
WYnAn-IFn-l
=
O.
KarlGustav Jacob Jacobi (1804-1851) was the secoudson
born
10 a well-to-doJewish baoking family in Potsdam. An
obviously bright young man, Jacobi was soon moved to the
highest class in spite of his youth and remained at the gym-
nasium for four years only because he could not enter the
university until he was sixteen. He excelled at the University
of
Berlinin all the classical subjects as well as mathematical
studies,the topic he soon chose as his career.He passed the
examinationtobecomea secondaryschoolteacher,then later
the examination that allowed university teaching, and joined
the faculty at Berlin at the age
of
twenty. Since promotion
there appearedunlikely,he movedin 1826 to the Universityof Konigsberg in search of a
more permanentposition. He was
known
as a lively and creative lecturer who often injected
his latest research topics into the lectures. He began what is now a common practice at
most
universities-the
research
seminar-for
the
most
advanced students and his faculty
collaborators. The Jacobi "school,"togetherwiththe influence
of
Bessel and Neumann(also
at Konigsberg), sparked a renewal
of
mathematical excellence in Germany.
In
1843 Jacobi fell gravely
ill
with diabetes. After seeing his condition,
Dirichlet,
with
the help
of
von
Humboldt,
secured a donation to enable Jacobi to spend several months in
Italy, a therapy recommendedby his doctor. The friendly atmosphere and healthful climate
there soon improved his condition. Jacobi was later given royal permission to move from
Konigsberg10 Berlin so that his heal!h would not be affectedby the harsh winlers in the
former location. A salary bonusgiven to Jacobito offset the
highercostofliving
in the capital
was revoked after he made some politically sensitive remarks in an impromptu speech. A
permanentposition
at Berlin was also refused, and the reduced salary and lack of security
caused considerablehardship for Jacobi and his family. Only afterhe accepteda positionin
Vienna did the Prussian governmentrecognize the desirability
of
keeping the distinguished
mathematician within its borders, offering
him
special concessions that together with his
love for his homeland convinced Jacobi to stay.
In 1851 Jacobi died after contracting both
influenza and smallpox.
Jacobi'smathematicalreputationbeganlargelywith his heated competition with Abelin
the study
of
ellipticfunctions. Legendre, formerly the star
of
such studies,wroteJacobi
of
his
happiness at having"livedlong enoughto witnessthese magnanimous contestsbetweentwo
young athletes equally strong." Although Jacobi and Abel couldreasonably be considered
contemporary researchers who arrived at many
of
the same results independently, Jacobi
suggested the names "Abelian functions" and "Abelian theorem" in a review he wrote for
Crelle's Journal. Jacobi also extendedhis discoveries in elliptic functions to numbertheory
and the theory
of
integration. He also worked in other areas
of
number theory, such as the
theory
of
quadraticforms and the representation
of
integers assums
of
squaresand cubes. He
(7.6)
(7.7)
178 7.
CLASSICAL
ORTHOGONAL
POLYNOMIALS
presented
thewell-knownJacobian, orfunctional
determinant,
in1841.Tophysicists,
Jacobi
is
probably
best
known
forhis workin
dynamics
withthe
form
introduced
by
Hamilton.
Althoughelegantand
quite
general,
Hamiltonian
dynamics
did notlenditselfto easysolution
of
many
practical
problems
in
mechanics.
In the
spirit
of
Lagrange,
Poisson,
and
others,
Jacobi
investigated
transformations
of
Hamilton's
equations
that
preserved
their
canonical
nature
(loosely
speaking,
that
preserved
thePoisson
brackets
ineach
representation).
After
much
work
anda little
simplification,
the
resulting
equations
of
motion,
now
known
as
Hamilton-Jacobi equations, allowed
Jacobi
tosolve
several
important
problems
in
ordinary
andcelestial
mechanics.
Clebsch
and
later
Helmholtz
amplified
their
use in
other
areas
of
physics.
We can get another recurrence relation involving derivatives by substituting
(7.4) in (7.5) and simplifying:
ZwsanF~
+[an
:X
(ws) +wO'
n
- An+t)(anx +
fJn)]
F
n
+wYn(An-l - An+l)Fn-l =
O.
Two other recurrence relations can be obtained by differentiating Equations
(7.6) and (7.5), respectively, and using the differential equation for
F
n
•
Now solve
the first equationso obtained for
Yn(djdx)(wFn-l)
and substitote the result in the
second equation. After simplification, the result will be
ZwanAnFn +
:x
{[an
:x
(ws) +W(A
n
-
An-l)(anx
+
fJn)]
F
n}
d
+(An-I - An+t)
dx
(WFn+l) = O.
Finally, we record one more useful recurrence relation:
dw dw
An(x)F
n
- An+l(anx +
fJn)-d
Fn+t +YnAn-l(anx +
fJn)-Fn-1
x
dx
+
Bn(x)F~+t
+
YnDn(x)F~_1
= 0, (7.8)
where
An
(x)
= (anx +
fJn)
[zwanA
n
+an d
2
2
(ws) +An(anx +
fJn)
dW]
dx
dx
2 d
- an
dx
(ws),
d
Bn(x) = an
dx
(ws) -
w(anx
+
fJn)(An+1
- An),
d
Dn(x) =
w(anx
+fJn)(An-l - An) - an
-(ws).
dx
Details of the derivation
of
this relation are left for the reader. All these recurrence
relations seem to be very complicated. However, complexity is the price we pay
7.4
EXAMPLES
OF
CLASSICAL
ORTHOGONAL
POLYNOMIALS
179
for generality. When we work with specific orthogonal polynomials, the equa-
tions simplify considerably. For instance, for Hermite and Legendre polynomials
Equation (7.6) yields, respectively,
useful
recurrence
relations
for
Hermite
and
Legendre
polynomials
and
2 '
(l
- X
)P
n
+nxp" -
nPn-l
=
O.
(7.9)
Also, applying Equation (7.7) to Legendre polynomials gives
, ,
Pn+1
- XP
n
- (n +
I)P
n
= 0,
and Equation (7.8) yields
, ,
P
n
+
1
- P
n-
1
- (2n
+
I)P
n
=
O.
(7.10)
(7.11)
It
is possible to find many more recurrence relations by manipulating the ex-
isting
recurrence
relations.
Before studying specific orthogonal polynomials, let us pause for a moment
to appreciate the generality and elegance
of
the foregoing discussion. With a few
assumptions and a single defining equation we have severely restricted the choice
of
the weight function and with it the choice
of
the interval (a, b). We have nev-
ertheless exhausted the list
of
the so-called classical orthogonal polynomials.
7.4 Examples
of
Classical Orthogonal Polynomials
We now consttuct the specific polynomials used frequently in physics. We have
seen thatthe four parameters
K
n,
k
n,
k~,
and h
n
determine all the properties
of
the
polynomials. Once
K
n
is fixed by some standardization, we can detenmine all the
other parameters:
k
n
and
k~
will be given by the generalized Rodriguez formula,
and
h
n
can be calculated as
h
n
= l
b
F;(x)w(x)
dx
=l
b
(knx
n
+...)Fn(x)W(X)
dx
l
b
I d
n
k l
b
d
[d
n-
1
]
=k
n
wxn----(wsn)dx
= -'0. x
n_
--I
(ws
n)
dx
a Knw
dx"
K
n
a dx
dx»:
k d
n-
1
[b
k l
b
d d
n-
I
= -'0. xn
__
(ws
n)
_ -'0.
_(xn)
__
(ws
n)
dx
K
n
dxr:"
a K
n
a
dx dx
n-1
The first term
of
the last line is zero by
Lemma
7.1.3.
It
is clear that eachintegra-
tion by parts introduces a minus
sign and shifts
one
differentiation from uis" to
x".Thus, after n integrations by parts and noting that
dO
j
dx
o(
ws
n)
= ius" and
d"
jdxn(x
n)
= n!, we obtain
h
(-l)nknn!
l
b
nd
n = WS x.
K
n
a
(7.12)
180 7.
CLASSICAL
ORTHOGONAL
POLYNOMIALS
summary
of
properties
of
Hermite
polynomials
7.4.1 HermitePolynomials
The
Hermite
polynomials are standardized such that K
n
=
(_1)n.
Thus, the
geueralized Rodriguez formula (7.2) and Proposition 7.2.1 give
2 d" 2
Hn(x) =
(_I)"e
x
-
(e-
X
).
dx
n
(7.13)
It
is clear that each time
e-
x2
is differentiated, a factor
of
-2x
is introduced.
The highest
power
of
x is obtained
when
we differentiate
e-
x2
n
times. This yields
2 2
(_I)nex
(_2x)n
e
- x =2
nxn
'* k
n
=2
n.
To obtain
l(,.,
we find it helpful to see whether
the
polynomial is even or odd.
We substitute
-x
for x in Equation (7.13)
and
get
Hn(-x)
=
(_I)n
Hn(x), which
shows
thatifn
is even (odd), H
n
is an even (odd) polynomial, i.e., it
can
have
only
even (odd) powers
of
x.
In
either
case, the next-highest
power
of
x in Hn(x) is
not
n - I but n - 2. Thus, the coefficient
of
x
n-
1
is zero for Hn(x),
and
we have
k~
=
O.
For
h«, we use (7.12) to obtain h
n
=
.,jii2
nnL
Next we calculate the recurrence relation
of
Equation (5.12). We
can
readily
calculate the constants needed: an
= 2, fin = 0, Yn =
-2n.
Then
substitute these
in Equation (5.12) to obtain
(7.14)
(7.15)
(7.16)
summary
of
properties
of
Laguerre
polynomiais
Other
recurrence relations
can
be obtained similarly.
Finally,
the
differential equation
of
H
n
(x)
is obtained by first noting that K1 =
-I,
(jz
= 0, Fl (x) =2x '* kl = 2.
All
of
this gives An =
-2n,
which
can
be
used
in
the
equation
of
Proposition 7.1.4 to
get
dZH
n
an;
dx
z
-
2x
dx
+
2nH
n
=
O.
7.4.2 LaguerrePolynomials
For
Laguerrepolynomials,
the
standardization is K
n
= nL Thus, the generalized
Rodriguez formula (7.2) and Proposition 7.2.1 give
I d
n
I d
n
L~(x)
=
(xVe-xx
n)
=
_x-VeX_(xn+ve-X).
n!xVe
x
dx"
n!
dx"
To find k
n
we note thatdifferentiating
e-
x
does
not
introduce any
new
powers
of
x but ouly a factor
of
-1.
Thus,
the
highest
power
of
x is obtained by leaving
x
n
+
v
alone
and
differentiating
e-
x
n times.
This
gives
~,x-vexXn+V(_I)ne-X
=
(_I)n
xn '* k
n
=
(_I)n.
n. n! n!
7.4
EXAMPLES
OF
CLASSICAL
ORTHOGONAL
POLYNOMIALS
181
We may try to check the evenness or oddness
of
L~
(x);
however, this will not
be helpful because changing
x to
-x
distorts the RHS of Equation (7.16). In fact,
k~
i'
0 in this case, and it can be calculatedby noticing that the next-highestpower
of
x is obtained by adding the first derivative of x"+v n times and multiplying the
result by
(_1)n-l
, which comes from differentiating
e-
x.
We obtain
I (
1)"-I(n
+v)
-v
X[(
1)"-1
( + )
n+v-l
-X]
- ,,-1
-x
e - n n v x e =
)1
X ,
n! (n
-I
.
and therefore
k~
=
(-I)"-I(n
+
v)/(n
- I)L
Finally, for h" we
get
h
_
(-1)"[(-1)"
/n!]n!
1""
v
-x
"d
_ I
1""
n+v
-x
d
n - x e x x - - x e x.
n! 0 n! 0
If
v is not an integer (and it need not be), the integral on the RHS cannot be
evaluated by elementary methods.
In fact, this integral occurs so frequently in
the
gamma
function
mathematical applications that it is given a special name, the
gamma
function.
A detailed discussion of this function
can
be found in Chapter 11.
At
this point,
we simply note that
['en +1) = n! for n
EN,
(7.17)
and write h
n
as
h
n
= ['en +v +1) = ['en +v +1).
n! ['en +1)
The relevant parameters for the recurrence relation
can
be easily calculated:
1
Ol
----
,,-
n+I'
fJ
_
2n+v+I
n-
n+l
'
n+v
Y"=-n+I'
Substituting these in Equation (5.12) and simplifying yields
(n +
I)L~+l
= (2n +v + I -
x)L~
- (n +
v)L~_I'
With
kl
=
-I
and (f2 = 0, we get
A"
=
-n,
and the differential equation of
Proposition
7.1.4 becomes
d
2
p
ai:
x--f
+(v +
I-x)--"
+nL~
=
O.
dx dx
(7.18)
182 7.
CLASSICAL
ORTHOGONAL
POLYNOMIALS
7.4.3 Legendre Polynomials
summary
of
properties
of
Legendre
polynomials
Instead
of
discussing the Jacobi polynomials as a whole, we will discuss a special
case
of
them, the Legendre polynomials P"
(x),
which are more widely used in
physics.
With
IJ.
= 0 = v, corresponding to the Legendre polynomials, the weight
function for the Jacobi polynomials reduces to
w(x)
= I. The standardization is
K" =
(-
1)"2"n!.Thus, the generalized Rodriguez formula reads
(_I)"
a"
2
P,,(x) =
----
[(1-
x)"].
2
n
n!
dx"
(7.19)
Tofind
k", we expandthe expressionin squarebracketsusing the binomialtheorem
and take the
nth
derivative
of
the highest power
of
x. This yields
k x" _
(_I)"
a"
_x2)"
__
1_
a"
(x2n)
" - 2" , a
II
[(
] - 2" I a "
n. x n. x
=
_1_
2n(2n
- 1)(2n -
2)
...
(n +
I)x".
2
n
n!
2"r(n
+
1)
After some algebra (see Problem 7.7), we get k" = 1 2
n!r(z)
Adrien-Marie
Legendre
(1752-1833) came from a well-
to-do Parisian family
and
received an excellent education in
science
and
mathematics. His university
work
was advanced
enough that his mentor used many of Legendre's essays in
a treatise on mechanics. A man of modest fortune until the
revolution, Legendre was able to devote himself to study
and research without recourse to an academic position.
In
1782 he won the prize of the Berlin Academy for calculat-
ing the trajectories
of
cannonballs taking air resistance into
account. This essay broughthim to the attention
of
Lagrange
and helped pave the way to acceptance in French scientific
circles, notably the Academy of Sciences, to which Legendre submitted numerous papers.
In July 1784 he submitted a paperon planetary orbits that contained the now-famous Leg-
endre polynomials,
mentioning that Lagrange had been able to "present a more complete
theory" in a recentpaperby using Legendre's results.
In the years that followed, Legendre
concentrated his efforts in number theory, celestial mechanics, and the theory of elliptic
functions.
In addition, he was a prolific calculator, producing large tables of the values of
special functions, and he also authored an elementary textbook_that remained
in use for
many decades.
In 1824 Legendre refused to vote for the government's candidate for Institut
National.
Because of this, his pension was stopped and he died in poverty and in pain at the
age of 80 after several years of failing health.
Legendre produced a large number of useful ideas but did not always develop them
in the most rigorous manner, claiming to hold the priority for an idea
if
he had presented