The document discusses Legendre functions, which are solutions to Legendre's differential equation. Legendre functions arise when solving Laplace's equation in spherical coordinates. Legendre polynomials were first introduced by Adrien-Marie Legendre in 1785 as coefficients in an expansion of Newtonian potential. The document covers topics such as Legendre polynomials, Rodrigues' formula, orthogonality of Legendre polynomials, and associated Legendre functions.
2. Table of Content
SOLO
2
Legendre Functions
Introduction
Legendre Polynomials History
Second Order Linear Ordinary Differential Equation (ODE)
Laplace’s Homogeneous Differential Equation
Legendre Polynomials
The Generating Function of Legendre Polynomials
Rodrigues' Formula
Series Solutions – Frobenius’ Method
Recursive Relations for Legendre Polynomial Computation
Orthogonality of Legendre Polynomials
Expansion of Functions, Legendre Series
Schlaefli Integral
Laplace’s Integral Representation
Neumann Integral
3. Table of Content (continue)
SOLO
3
Legendre Functions
Associate Lagrange Differential Equation
Laplace Differential Equation in Spherical Coordinates
Analysis of Associate Lagrange Differential Equation
Associate Lagrange Differential Equation (2nd
Way)
Generating Function for Pn
m
(t)
Recurrence Relations for Pn
m
(t)
Orthogonality of Associated Legendre Functions
Recurrence Relations for Θn
m
Functions
Spherical Harmonics
References
Boundary Value Problems and Sturm–Liouville Theory
4. SOLO
4
Legendre Polynomials
Adrien-Marie Legendre
(1752 –1833(
In mathematics, Legendre functions are solutions to Legendre's
differential equation:
( ) ( ) ( ) ( ) 011 2
=++
− xPnnxP
xd
d
x
xd
d
nn
They are named after Adrien-Marie Legendre. This
ordinary differential equation is frequently encountered in
physics and other technical fields. In particular, it occurs
when solving Laplace's equation (and related partial
differential equations) in spherical coordinates.
The Legendre polynomials were first introduced in 1785 by Adrien-Marie
Legendre, in “Recherches sur l’attraction des sphéroides homogènes”, as the
coefficients in the expansion of the Newtonian potential
( )∑
∞
=
=
−
+
=
−+
=
−
0
222
cos
'1
cos
'
2
'
1
11
cos'2'
1
'
1
n
n
n
P
r
r
r
r
r
r
rrrrrrrr
γ
γ
γ
Return to Table of Content
5. Second Order Linear Ordinary Differential Equation (ODE)
Consider a Second Order Linear Ordinary Differential Equation (ODE) define by
the Operator
SOLO
( ) ( ) ( ) ( ) ( )xuxp
xd
ud
xp
xd
ud
xpxu 212
2
0: ++=L
defined in the interval a ≤ x ≤ b, with the coefficients p0 (x), p1 (x), p2 (x) real in
this region and with the first 2 – i derivatives of pi (x) continuous . Also we
require that p0 (x) is nonzero in the interval.
Define the Quadratic Form of the Operator L as:
( ) ( ) ( ) dxuup
xd
ud
p
xd
ud
pdxxuxuxu
b
a
b
a
∫∫
++== 212
2
0: LLu,
( ) ( ) ( )
( ) ( ) ( ) ( )∫ ∫∫
∫ ∫∫
∫∫∫
+−++
−
=
+−+−
=
++
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
xdupuxdup
xd
d
uupudxup
xd
d
uup
xd
d
u
xd
ud
up
xdupuxdup
xd
d
uupudx
xd
ud
up
xd
d
xd
ud
up
xdupxdu
xd
ud
pxdu
xd
ud
p
21102
2
00
21100
2
212
2
0
5
6. SOLO
( ) ( ) ( ) dxuup
xd
ud
p
xd
ud
pdxxuxuxu
b
a
b
a
∫∫
++== 212
2
0: LLu,
( ) ( )
( ) ( )
b
a
b
a
b
a
b
a
b
a
xu
xd
pd
pxu
upu
xd
ud
upu
xd
pd
u
xd
ud
upupuup
xd
d
u
xd
ud
up
−=
+−−=+
−
0
1
10
0
0100
Therefore:
( ) ( ) ( ) ( )∫
+−+
−=
b
a
b
a
dxupup
xd
d
up
xd
d
uxu
xd
pd
pxu 2102
2
0
1
Define the Adjoint Operator:
( ) ( ) ( ) upup
xd
d
up
xd
d
xu 2102
2
: +−=L
6
Second Order Linear Ordinary Differential Equation (ODE)
7. SOLO
( ) dx
u
up
xd
ud
p
xd
ud
puxu
b
a
∫
++=
L
Lu, 212
2
0
The two Integrals are equal if:
( ) ( ) ( ) ( )∫
+−+
−=
b
a
b
a
dxupup
xd
d
up
xd
d
uxu
xd
pd
pxu
uL
2102
2
0
1
or:
( ) ( ) ( ) ( ) ( ) ( )xuxq
xd
xud
xp
xd
d
xuxu +
== LL
( ) ( )
02 1
01
2
0
2
2102
2
212
2
0
=
−+
−=
+−−
++
xd
ud
p
xd
pd
uu
xd
pd
xd
pd
u
upup
xd
d
up
xd
d
uup
xd
ud
p
xd
ud
pu
( ) ( )
xd
xpd
xp 0
1 =
If this condition is satisfied, then the terms at the boundary x = a and x = b also
vanish, and we have by defining p (x): = p0 (x) and q (x) := p2 (x)
7
Second Order Linear Ordinary Differential Equation (ODE)
8. SOLO
( ) ( ) ( ) ( ) ( ) ( )xuxq
xd
xud
xp
xd
d
xuxu +
== LL
This Operator is called Self-Adjoint
( )
( )
( )
( )
( )
( )
( )
( ) ( ) ( ) ( )
++
=
∫∫ xuxp
xd
ud
xp
xd
ud
xptd
tp
tp
xp
xutd
tp
tp
xp xx
212
2
0
0
1
00
1
0
expexp
1
L
1
( )
( )
( )
( )
( )
( )
( )
( )
( )xutd
tp
tp
xp
xp
xd
ud
td
tp
tp
xd
d
xq
x
xp
x
+
= ∫∫ 0
1
0
2
0
1
expexp This is clearly Self-Adjoint.
We can see that p0 (x) is in the denominator. This is the reason that we requested the
p0 (x) to be nonzero in the interval a ≤ x ≤ b. p0 (x1) = 0 means that the Differential
Equation is not Second Order at that point.
We can always transform the Non-Self-Adjoint Operator to a Self-Adjoint one
by multipling L by
( )
( )
( )
∫x
td
tp
tp
xp 0
1
0
exp
1
8
Second Order Linear Ordinary Differential Equation (ODE)
9. Consider a Second Order Linear Ordinary Differential Equation (ODE)
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
2
210 :'':'0''':
xd
ud
u
xd
ud
uxuxpxuxpxuxpxu ===++=L
If we know one Solution u1 (x) we can find a second u2 (x).
Proof
If we have two Solutions u1 (x) and u2 (x), then
0'''
0'''
222120
121110
=++
=++
upupup
upupup
Multiply first equation by u2 (x) and second by u1 (x) and subtract:
( ) ( ) 0'''''' 1221112210 =−+− uuuupuuuup
The Wronskian W is defined as:
1221
21
21
''
''
: uuuu
uu
uu
W −==
0' 10 =+ WpWp
9
Theorem: “Method of Reduction of Order”
Second Order Linear Ordinary Differential Equation (ODE)
10. Consider a Second Order Linear Ordinary Differential Equation (ODE)
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
2
210 :'':'0''':
xd
ud
u
xd
ud
uxuxpxuxpxuxpxu ===++=L
Theorem: “Method of Reduction of Order”
If we know one Solution u1 (x) we can find a second u2 (x).
Proof (continue -1)
0' 10 =+ WpWp xd
p
p
W
Wd
0
1
−= ( ) ( ) ( )
( )
−= ∫
x
x
d
p
p
xWxW
0 0
1
0 exp τ
τ
τ
From: ( ) ( ) ( ) ( ) ( )xuxuxuxuxW 1221 '' −=
q.e.d.
Therefore: ( ) ( ) ( )
( )∫=
x
x
d
u
W
xuxu
0
2
1
12 τ
τ
τ
( )
( )
( )
( )
( )
( )
( )
=−=
1
2
22
1
1
1
2
2
1
''
u
u
xd
d
xu
xu
xu
xu
xu
xu
xW
Divide by u1
2
(x)
x0 and W (x0) are given (or not).
10
Second Order Linear Ordinary Differential Equation (ODE)
11. If the Second Order Linear Ordinary Differential Equation (ODE) is in the
Self-Adjoint Mode:
SOLO
Then:
The Wronskian is given by
( ) ( ) ( ) ( ) ( ) ( ) 0=+
== xuxq
xd
xud
xp
xd
d
xuxu LL
( )
( ) ( ) ( ) ( ) ( ) 0
1
0
2
2
=++ xuxq
xd
xud
xd
xpd
xd
xud
xp
p
p
( ) ( ) ( )
( )
( ) ( )
( )
( )xp
xp
xW
p
pd
xWd
p
p
xWxW
x
x
x
x
0
00
0
0
0
0
0
1
0
00
expexp =
−=
−= ∫∫ τ
τ
τ
11
Return to Table of Content
Second Order Linear Ordinary Differential Equation (ODE)
12. 12
SOLO
The Laplace’s Homogeneous Differential Equation is:
We want to find the Potential Φ at the point F (field) due to all the sources (S) in the
volume V, including its boundaries .
n
i
iSS
1=
=
iS
nS
n
i
iSS
1=
=dV
dSn
→
1
V
Fr
Sr
F
0r
SF rrr
−= iS
nS
dV
dSn
→
1
V
Fr
Sr
F
0r SF rrr
−=
F inside V F on the boundary of V
Therefore is the vector from S to F.SF rrr
−=
→→→
++= zzyyxxr 111
→→→
++= zzyyxxr SSSS 111
→→→
++= zzyyxxr FFFF 111
Let define the operator
that acts only on the
source coordinate.
→→→
∂
∂
+
∂
∂
+
∂
∂
=∇ z
z
y
y
x
x SSS
S 111
Sr
Laplace’s Homogeneous Differential Equation
( ) 02
=Φ∇ r
Pierre-Simon,
marquis de Laplace
(1749 1827)
13. 13
SOLO
Laplace’s Homogeneous Differential Equation
Laplace Differential Equation in Spherical Coordinates
( ) 02
=Φ∇ r
Spherical Coordinates:
θ
ϕθ
ϕθ
cos
sinsin
cossin
rz
ry
rx
=
=
=
A Solution in Spherical Coordinates is: ( ) 0
1
≠=Φ r
r
r
( ) 0
111
22
≠−=∇−=∇=Φ∇ r
r
r
r
r
rr
r
( ) ( ) 00
33111
35333
2
≠=−⋅=⋅∇−⋅
−∇=
⋅−∇=Φ∇⋅∇=Φ∇ r
r
rr
r
r
r
r
r
r
r
rr
r
r
r
z
r
y
r
x
zyx
zyx
r zyxzyx
=++=++
∂
∂
+
∂
∂
+
∂
∂
=∇ 111111 222
( ) 3111111 =++⋅
∂
∂
+
∂
∂
+
∂
∂
=⋅∇ zyx
zyx
r zyxzyx
2222
zyxr ++=
14. 14
SOLO
Laplace’s Homogeneous Differential Equation
Laplace Differential Equation in Spherical Coordinates
Spherical Coordinates:
θ
ϕθ
ϕθ
cos
sinsin
cossin
rz
ry
rx
=
=
=
θcos=
∂
∂
z
r
2222
zyxr ++=
0
cos11
22
≠−=
∂
∂
−=
∂
∂
=
∂
Φ∂
r
rz
r
rrzz
θ
0
1cos3331
3
2
5
22
4332
2
≠
−
=
−
=
∂
∂
+−=
−
∂
∂
=
∂
Φ∂
r
rr
rz
z
r
r
z
rr
z
zz
θ
0
cos9cos159153
5
26
3
4
23
7
23
6
22
55
22
3
3
≠
−
=
−
−=
∂
∂−
−
∂
∂
−
=
−
∂
∂
=
∂
Φ∂
r
r
r
r
rzz
z
r
r
rz
r
z
r
rz
r
rz
zz
θθ
We note that ∂n
Φ/∂zn
is a n-degree polynomial in cos θ divided by rn+1
.
( ) ( ) ( ) ( ) ( ) ( )zyx
zn
hzyx
z
hzyx
z
hzyxhzyx n
nn
n
,,
!
1
,,
!2
1
,,,,,, 2
2
2
∂
Φ∂−
++
∂
Φ∂
+
∂
Φ∂
−Φ=−Φ
Using Taylor’s Series development we obtain
15. 15
SOLO
Laplace’s Homogeneous Differential Equation
Laplace Differential Equation in Spherical Coordinates
Spherical Coordinates:
θ
ϕθ
ϕθ
cos
sinsin
cossin
rz
ry
rx
=
=
=
2222
zyxr ++=
( )
( )
( )
∂
∂−
++
∂
∂
+
∂
∂
−=
−++
=−Φ
rzn
h
rz
h
rz
h
rhzyx
hzyx n
nn
n 1
!
11
!2
1111
,, 2
2
2
222
( )
∑
∞
=
∂
∂−
=
+− 0
22
1
!
1
cos2
1
n
n
nn
n
rzn
h
hhrr θ
or:
Let define:
( ) ( )
∂
∂−
= +
rzn
rP n
nn
n
n
1
!
1
:cos 1
θ
( ) rhP
r
h
rhhrr n
n
n
≤
=
+−
∑
∞
=0
22
cos
1
cos2
1
θ
θ
Therefore:
16. 16
SOLO
Laplace’s Homogeneous Differential Equation
Laplace Differential Equation in Spherical Coordinates
Spherical Coordinates:
θ
ϕθ
ϕθ
cos
sinsin
cossin
rz
ry
rx
=
=
=
2222
zyxr ++=
Let define w:=Pn (cos θ) and substitute w/rn+1
in the
Laplace Equations in Spherical Coordinates instead of Φ
Therefore, since w:=Pn (cos θ) (Partial Derivative becomes Total Derivative):
0
sin
1
sin
sin
11
0
12
2
22121
2
2
=
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+++
nnn
r
w
rr
w
rr
w
r
r
rr φθθ
θ
θθ
0sin
sin
111
1
=
∂
∂
∂
∂
+
+
−
∂
∂
+
θ
θ
θθ
w
rr
n
r
w nn
or:
( ) 0sin
sin
1
1 =
++
θ
θ
θθ d
wd
d
d
wnn
Substitute t=cos θ (dt = - sinθ dθ):
( ) ( )
−=
−−=
=
−−
td
wd
t
td
d
td
wd
t
td
d
d
td
td
wd
d
td
d
d
d
wd
d
d 2
sin
2
1sin
11
sin
1
sin
sin
1
sin
sin
1
2
θ
θ
θθθ
θ
θθθ
θ
θθ
Finally we obtain: ( ) ( ) 1011 2
≤=++
− twnn
td
wd
t
td
d
Return to Table of Content
17. 17
SOLO
This is the Legendre Differential Equation and Pn (t) the Legendre Polynomials
are one of the two solutions of the ODE.
( ) ( ) ( ) ( ) 1011 2
≤=++
− ttPnntP
td
d
t
td
d
nn
( )∑
∞
=
=
+−
0
2
cos
cos21
1
n
n
n
P
r
h
r
h
r
h
θ
θ
We found
( ) 1cos
cos21
1
0
2
≤=
+−
∑
∞
=
uPu
uu n
n
n
θ
θFor u:=h/r
The left-hand side is called “The Generating Function of Legendre
Polynomials”
Legendre Polynomials
The Generating Function of Legendre Polynomials
18. 18
SOLO
Let use Taylor expansion
for the function:
( ) ( )
( )
( ) ( )
( ) ( )
( ) +++++= n
n
x
n
f
x
f
x
f
fxf
!
0
!2
0
1
0
0 2
21
( )
( ) 1,
21
1
0
2
≤=
−+
∑
∞
=
tutPu
tuu n
n
n
( ) ( ) ( ) 101 2
1
=−=
−
fxxf
( )
( ) ( ) ( )
( )
2
1
01
2
1 1
2
3
1
=−=
−
fxxf
( )
( ) ( ) ( )
( )
2
3
2
1
01
2
3
2
1 2
2
5
2
⋅=−⋅=
−
fxxf
( )
( ) ( ) ( )
( ) ( ) ( )
!2
!2
!2
!2
2
1231
2
12
2
3
2
1
01
2
12
2
3
2
1
2
2
12
n
n
n
nnn
fx
n
xf nn
n
n
n
n
n
=⋅
−⋅
=
−
⋅=−
−
⋅=
+
−
Tacking x = 2 u t - u2
we obtain
( )
( )
( )
( ) 12
!2
!2
21
1
0
2
222
≤−=
−−
∑
∞
=
uutu
n
n
utu n
n
n
Let prove that Pn (t) are indeed Polynomials.
Legendre Polynomials
The Generating Function of Legendre Polynomials
19. 19
SOLO
Using Taylor expansion we obtained:
( )
( ) 1,
21
1
0
2
≤=
−+
∑
∞
=
tutPu
tuu n
n
n
Take the binomial expansion of (2 u t - u2
)n
we obtain
( )
( )
( )
( ) 12
!2
!2
21
1
0
2
222
≤−=
−−
∑
∞
=
uutu
n
n
utu n
n
n
( )
( )
( )
( )
( )
( ) ( ) ( )
( )
( ) 12
!!!2
!2
12
!!
!
1
!2
!2
21
1
0 0
2
0 0
222
≤
−
−=
−
−=
−−
∑∑∑ ∑
∞
= =
+−
∞
= =
−
uut
knkn
n
ut
knk
n
n
n
utu n
n
k
knkn
n
k
n
n
k
kknk
n
Change Variables in the second sum from n+k to n
( )
( ) ( )
( ) ( )
( )
[ ]
( )
−
=
≤
−−
−
−=
−−
∑ ∑
∞
= =
−
−
evennifn
oddnifnn
uut
knknk
kn
utu n
n
k
nkn
kn
k
2/1
2/
2
12
!2!!2
!22
1
21
1
0
2/
0
2
222
Equating in the two power series the un
coefficients, we obtain
( ) ( ) ( )
( ) ( )
[ ]
∑=
−
≤
−−
−
−=
2/
0
2
1
!2!!2
!22
1
n
k
kn
n
k
n tt
knknk
kn
tP
Polynomial
of order n in t
Return to Rodrtgues Formula
Return to Frobenius Series
We can see that for n odd the polynomial Pn (t) has only odd powers of t and
for n even only even powers of t.
Legendre Polynomials
The Generating Function of Legendre Polynomials
20. 20
SOLO
Let use Taylor expansion
for the function:
( ) ( )
( )
( ) ( )
( ) ( )
( ) +++++= n
n
x
n
f
x
f
x
f
fxf
!
0
!2
0
1
0
0 2
21
( )
( ) 1,
21
1
0
2
≤=
−+
∑
∞
=
tutPu
tuu n
n
n
Tacking x = u2
-2 u t we obtain
( ) ( ) ( ) 101 2
1
=+=
−
fxxf
( )
( ) ( ) ( )
( )
2
1
01
2
1 1
2
3
1
−=+−=
−
fxxf
( )
( ) ( ) ( )
( )
2
3
2
1
01
2
3
2
1 2
2
5
2
⋅=+⋅=
−
fxxf
( )
( ) ( ) ( ) ( )
( ) ( )
2
12
2
3
2
1
101
2
12
2
3
2
1
1 2
12 −
⋅−=+
−
⋅−=
+
− n
fx
n
xf
nn
n
nn
( )
( ) ( ) ( ) ( )
( )∑
∞
=
=
+−
+
−
+
−
++=
−+−−−+−−=
−+
0
24
4
3
3
2
20
4232222
2
8
33035
2
35
2
13
1
2
128
35
2
16
5
2
8
3
2
1
1
21
1
n
n
n
tPu
tt
u
tt
u
t
utuu
tuutuutuutuu
ttuu
Legendre Polynomials
The Generating Function of Legendre Polynomials
21. SOLO
21
Legendre Polynomials
The first few Legendre polynomials are:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) 256/63346530030900901093954618910
128/31546201801825740121559
128/35126069301201264358
16/353156934297
16/51053152316
8/1570635
8/330354
2/353
2/132
1
10
246810
3579
2468
357
246
35
24
3
2
−+++−
+−+−
+−+−
−+−
−+−
+−
+−
−
−
xxxxx
xxxxx
xxxx
xxxx
xxx
xxx
xx
xx
x
x
xPn n
22. 22
SOLO
( ) 1,
21
1
0
2
≤=
+−
∑
∞
=
tutPu
utu n
n
n
Substitute u by – u in this equation:
( ) ( ) ( ) 11
21
1
00
2
≤−=−=
++
∑∑
∞
=
∞
=
utPutPu
utu n
n
n
n
n
nn
which results in the following identity: ( ) ( ) ( )tPtP n
n
n 1−=−
For t =1 we have
( ) 11
1
1
21
1
0
2
≤=
−
=
+−
∑
∞
=
uPu
uuu n
n
n
But 1
1
1
0
≤=
−
∑
∞
=
uu
u n
n
By equalizing the coefficients of un
in the two sums, we obtain:
( ) nPn ∀=11
and
( ) ( ) ( ) ( )n
n
n
n PP 1111 −=−=−
Legendre Polynomials
The Generating Function of Legendre Polynomials
23. 23
SOLO
( ) 1,
21
1
0
2
≤=
+−
∑
∞
=
tutPu
utu n
n
n
For t = 0 we obtain:
Legendre Polynomials
( ) 10
1
1
0
2
≤=
+
∑
∞
=
uPu
u n
n
n
( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )∑∑∑
∞
=
∞
=
∞
=
−
−=
⋅−⋅⋅−⋅⋅
−=
−⋅⋅
−=
+
−−−−
++
−−
+
−+=+=
+
0
22
2
0
1
2
0
2
22222/12
2
!2
!2
1
!2
222642
!2
12531
1
!2
12531
1
!
2/123/12/1
!2
3/12/1
2
1
11
1
1
n
n
n
n
n
nn
n
n
n
n
n
n
n
n
un
n
nn
n
un
n
un
t
n
n
ttu
u
Therefore we have:
( ) ( )
( )
( ) 10
!2
!2
1
1
1
00
22
2
2
≤=−=
+
∑∑
∞
=
∞
=
uPu
n
un
u n
n
n
n
n
n
n
By equating coefficients of un
on both sides we obtain:
( ) ( ) ( )
( )
( ) 00
!2
!2
10
12
222
=
−=
+n
n
n
n
P
n
n
P
Return to Table of Content
The Generating Function of Legendre Polynomials
24. Derivation of Legendre Polynomials via Rodrigues’ Formula
SOLO
24
Legendre Polynomials
Benjamin Olinde Rodrigues
(1794-1851)
In mathematics, Rodrigues' Formula (formerly called the Ivory–
Jacobi formula) is a formula for Legendre polynomials
independently introduced by Olinde Rodrigues (1816), Sir James
Ivory (1824) and Carl Gustav Jacobi (1827). The name
"Rodrigues’ formula" was introduced by Heine in 1878, after
Hermite pointed out in 1865 that Rodrigues was the first to
discover it, and is also used for generalizations to other orthogonal
polynomials. Askey (2005) describes the history of the Rodrigues’
Formula in detail.
Rodrigues stated his formula for Legendre polynomials Pn
Carl Gustav Jacob Jacobi
(1804 –1851)
( ) ( )[ ]n
n
n
nn x
xd
d
n
xP 1
!2
1 2
−=
25. SOLO
25
Legendre Polynomials
Olinde Rodrigues
(1794-1851)
Start from the function: ( ) .12
constkxky
n
=−=
( ) 12
12:'
−
−==
n
xxkn
xd
yd
y
( ) ( ) ( ) 22212
2
2
11412:''
−−
−−+−==
nn
xxnknxkn
xd
yd
y
Let compute:
( ) ( ) ( ) ( ) ( ) '12211412''1
12222
yxnynxxnknxknyx
nn
−+=−−+−=−
−
or: ( ) ( ) 02'12''12
=−−+− ynyxnyx
Let differentiate the last equation n times with respect to x:
( )[ ] ( ) ( ) ( ) ( )
( ) ( ) ''1''2''1
00''1
3
''1
2
''1
1
''1''1
2
2
1
1
2
3
3
0
2
3
3
2
2
2
2
2
1
1
222
y
xd
d
nny
xd
d
xny
xd
d
x
y
xd
d
x
xd
dn
y
xd
d
x
xd
dn
y
xd
d
x
xd
dn
y
xd
d
xyx
xd
d
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
−
−
−
−
−
−
−
−
−
−
−++−=
+−
+−
+−
+−=−
( ) ( ) ( ) ( )
+−=
+−=− −
−
−
−
''12'
1
'12'12 1
1
1
1
y
xd
d
ny
xd
d
xny
xd
d
x
xd
dn
y
xd
d
xnyx
xd
d
n n
n
n
n
n
n
n
n
n
n
Derivation of Legendre Polynomials via Rodrigues’ Formula
26. SOLO
26
Legendre Polynomials
Olinde Rodrigues
(1794-1851)
Start from the function: ( ) .12
constkxky
n
=−=
( ) ( ) 02'12''12
=−−+− ynyxnyxDifferentiate n times with
respect to x:
( ) ( ) ''1''2''1 2
2
1
1
2
y
xd
d
nny
xd
d
xny
xd
d
x n
n
n
n
n
n
−
−
−
−
−++−
( ) 02''12 1
1
=−
+−+ −
−
y
xd
d
ny
xd
d
ny
xd
d
xn n
n
n
n
n
n
Define: a Polynomial( ) ( )[ ]n
n
n
n
n
x
xd
d
k
xd
yd
xw 1: 2
−==
( ) ( ) ( )[ ] 02'121'2''12
=−+−+−++− wnwnwxnwnnwxnwx
( ) ( ) ( ) ( )[ ] 02121'2''12
=−−+−+−++− wnnnnnwxnxxnwx
( ) ( ) 01'2''12
=+−+− wnnwxwx
This is Legendre’s Differential Equation. We proved that one of the solutions
are Polynomials. We can rewrite this equation in a Sturm-Liouville Form:
( ) ( ) 0112
=+−
− wnnw
xd
d
x
xd
d
Derivation of Legendre Polynomials via Rodrigues’ Formula
27. SOLO
27
Legendre Polynomials
Olinde Rodrigues
(1794-1851)
Let find k such that:
by using the fact that Pn (1) = 1
( ) ( )[ ]n
n
n
n
n
n x
xd
d
k
xd
yd
xP 12
−==
( ) ( )[ ] ( ) ( ) ( ) ( )
−+=
+−=−= ∑>0
22
1!2111
i
inn
v
n
u
n
n
n
n
n
n
n xxaxnkxx
xd
d
kxk
xd
d
xP
!2
1
n
k n
=
We recover the Rodrigues Formula:
( ) ( )[ ]n
n
n
nn x
xd
d
n
xP 1
!2
1 2
−=
Let use Leibnitz’s Rule (Binomial Expansion for the n Derivative
of a Product - with u:=(x-1)n
and v:=(x+1)n
):
( )
( )
( ) udvudvdnvddu
nn
vddunvdu
vdud
mnm
n
vud
nnnnn
n
m
mnmn
+++
−
++=
−
=⋅
−−−
=
−
∑
1221
0
!2
1
!!
!
We have:
( )
( )
1!2
!2
1
1
1!20
12
0
21
00
==
+++
−
++==
=
−−−
nkudvudvdnvddu
nn
vddunvdukxP n
xn
nnnnn
n
n
We can see from this Formula that Pn (x) is indeed a Polynomial of Order n in t.
Derivation of Legendre Polynomials via Rodrigues’ Formula
28. SOLO
28
Legendre Polynomials
Olinde Rodrigues
(1794-1851)
Let find an explicit expression for Pn (x) from Rodrigues’ Formula:
( ) ( )[ ]n
n
n
nn
n
n x
xd
d
nxd
yd
xP 1
!2
1 2
−==
Start with:
( ) ( )
( )∑=
−
−
−
=−
n
m
mnmn
x
mnm
n
x
0
22
1
!!
!
1
( ) ( )[ ] ( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
+
=
−−
−
−=
−−
−=
+−−
−
−=−=
∑
∑
∑
=
−
−=
≥
−−
≥
−−
oddnn
evennn
p
x
knknk
kn
x
nm
m
mnm
xnmmm
mnm
n
n
x
xd
d
n
xP
p
k
knk
n
knm
n
pm
nmmn
n
n
pm
nmmn
n
n
n
n
nn
2/12
2/
!2!!
!22
1
2
1
!2
!2
!!
1
1
2
1
12122
!!
!
1
!2
1
1
!2
1
0
2
2
22
( ) ( )
<
≥+−−
=
2/0
2/121222
nm
nmnmmm
x
xd
d m
n
n
We recover the result obtained by the Generating Function of Legendre Polynomials
Return to Frobenius Series
Return to Table of Content
Derivation of Legendre Polynomials via Rodrigues’ Formula
30. SOLO
Example: Legendre Equation (continue – 1)
( ) ( ) ( ) ( ) ( )[ ] 0111
00
2
=+++−++−++ ∑∑
∞
=
+
∞
=
−+
λ
λ
λ
λ
λ
λ λλλλ kk
xkkrraxkka
Denote, in the first sum λ = j +2 and in the second sum λ = j, to obtain:
( ) ( )
( ) ( ) ( ) ( ) ( )[ ]{ } 01112
11
0
2
1
1
2
0
=+++−+++++++
++−
∑
∞
=
+
+
−−
j
jk
jj
kk
xjkjkrrajkjka
xkkaxkka
All the coefficients of xk+j
must be zero, therefore
( ) 001 00 ≠=− akka
( ) 011 =+kka
( ) ( ) ( ) ( ) ( )[ ] 011122 =+++−+++++++ jkjkrrajkjka jj
30
Ferdinand Georg
Frobenius
1849 - 1917
Series Solutions – Frobenius’ Method
Legendre Polynomials
31. SOLO
Example: Legendre Equation (continue – 2)
( ) ( ) ( )
( ) ( )
( )[ ] ( )
( ) ( )
,2,1,0
12
1
12
11
2 =
++++
++++−
−=
++++
+++−+
−=+ j
jkjk
jkrjkr
a
jkjk
jkjkrr
aa jjj
( ) 01 =−kk
( ) 011 =+kka
( ) ( ) ( ) ( ) ( )[ ] 011122 =+++−+++++++ jkjkrrajkjka jj
0&1.3
0&0.2
0&0.1
1
1
1
==
≠=
==
ak
ak
akThree possible
solutions
The equation that, k (k+1) = 0, comes from the coefficient of the lowest power
of x, and is called Indicial Equation. It has two solutions for k
k = 0 and k = 1
The equation
gives the recursive relation
31
Series Solutions – Frobenius’ Method
Legendre Polynomials
32. SOLO
Example: Legendre Equation (continue – 3)
( ) 1001: ==⇒=− korkkkEquationIndicial
1. Using k = 0 and a1 = 0 we obtain a series of even powers of x
( ) ( )
( ) ( )
( ) ( )
( ) ( )
,2,1,0
21
1
21
11
2 =
++
++−
−=
++
+−+
−=+ j
jj
jrjr
a
jj
jjrr
aa jjj
( ) ( ) ( ) ( ) ( ) ( )
+
++−
+
+
−== 42
0
!4
312
!2
1
1: x
rrrr
x
rr
axpxy reven
( ) 01231 ==== +naaa
The recurrence relation results in the following expression for the coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
,3,2,1
!2
123124222
1 02 =
−+++⋅−+−+−
−= ma
m
mrrrrrmrmr
a
m
m
32
( ) ( ) ( )
( ) ( )
( )[ ] ( )
( ) ( )
,2,1,0
12
1
12
11
2 =
++++
++++−
−=
++++
+++−+
−=+ j
jkjk
jkrjkr
a
jkjk
jkjkrr
aa jjj
( ) ∑
∞
=
= 0
2
2m
m
meven xaxy
Series Solutions – Frobenius’ Method
Legendre Polynomials
33. SOLO
Example: Legendre Equation (continue – 4)
( ) 1001: ==⇒=− korkkkEquationIndicial
3. Using k = 1 and a1 = 0 we obtain a series of odd powers of x
( ) ( )( )
( ) ( )
( )( )
( ) ( )
.2,1,0
32
21
32
211
2 =
++
++−−
−=
++
++−+
−=+ j
jj
jrjr
a
jj
jjrr
aa jjj
( ) ( ) ( )( ) ( ) ( )( ) ( )
+
++−−
+
+−
−== 53
0
!4
4213
!3
21
: x
rrrr
x
rr
xaxqxy rodd
( ) 01231 ==== +naaa
The recurrence relation results in the following expression for the coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
,3,2,1
!12
24213212
1 02 =
+
+++−+−+−
−= ma
m
mrrrrmrmr
a
m
m
33Since pr (x) and qr (x) are two linearly independent solutions of the 2nd
Order
Linear Lagrange ODE, the final solution is y = c1 pn (x) + c2 qn (x)
( ) ( ) ( )
( ) ( )
( )[ ] ( )
( ) ( )
,2,1,0
12
1
12
11
2 =
++++
++++−
−=
++++
+++−+
−=+ j
jkjk
jkrjkr
a
jkjk
jkjkrr
aa jjj
( ) ∑
∞
=
+
= 0
12
2m
m
modd xaxy
Series Solutions – Frobenius’ Method
Legendre Polynomials
34. SOLO
Example: Legendre Equation (continue – 5)
Using d’Alembert – Cauchy test for convergence of an Infinite Series, we have
Convergence Test:
( ) ( ) ( )
( ) ( )
≥≥
<<
=
++++
+++−+
=
∞→
+
+
∞→ divergex
convergex
xx
jkjk
jkjkrr
xa
xa
jj
j
j
j
j 11
11
12
11
limlim 22
2
2
The even series stops at j = n. The expansion is a Polynomial of order n (even).
1. If n is even, using k = 0 and a1 = 0 we obtain a series of even powers of x .
( ) ( )
( ) ( )
,..1,0
21
11
2 =
++
+−+
−=+ j
jj
jjnn
aa jj
For the case that r = n, a positive integer:
,...2,1,,02 ++==+ nnnjaj
3. If n is odd, using k = 1 and a1 = 0 we obtain a series of odd powers of x .
( ) ( )( )
( ) ( )
,..2,1
32
211
2 =
++
++−+
−=+ j
jj
jjnn
aa jj
,...2,1,,1,02 ++−==+ nnnnjaj
The odd series stops at j = n-1. The expansion is a Polynomial of order n (odd).34
Series Solutions – Frobenius’ Method
Legendre Polynomials
35. SOLO
Example: Legendre Equation (continue – 6)
1. If n is even, using k = 0 and a1 = 0 we obtain a series of even powers of x
( ) ( ) ( )( ) ( )
+
++−
+
+
−= 42
0
!4
312
!2
1
1 x
nnnn
x
nn
axyeven
3. If n is odd using k = 1 and a1 = 0 we obtain a series of odd powers of x
( ) ( )( ) ( ) ( )( ) ( )
+
++−−
+
+−
−= 53
0
!4
4231
!3
21
x
nnnn
x
nn
xaxyodd
( ) ( )
( ) ( )( ) ( )
+−=
++−
+
+
−==
−=
+
−==
==
42
0
42
0
2
0
2
0
0
6
70
101
!4
3414244
!2
144
14
31
!2
122
12
0
xxaxxayn
xaxayn
ayn
even
even
even
( )( )
−=
+−
−==
==
3
0
3
0
0
3
5
!3
2313
3
1
xxaxxayn
xayn
odd
odd
We obtain the Legendre Polynomials Solutions for a0 = 1. 35
Series Solutions – Frobenius’ Method
Legendre Polynomials
36. SOLO
Example: Legendre Equation (continue – 7)
1. The recurrence relation for even x powers results in the following expression for the
coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
2/,,3,2,1
!2
123124222
1 02 nma
m
mnnnnnmnmn
a
m
m
=
−+++⋅−+−+−
−=
( ) ( ) ( ) ( ) ( ) ( )
( )!
!
2121224222 11
2
mp
p
ppmpmpnnmnmn mm
pn
−
=−+−+−=−+−+− −−
=
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( )( ) ( )
( )
( ) ( )( ) ( )
( )
( ) ( )!!22
!!22
212
1
!2
!22
242
242
1231
!
!
1231
22
mpp
pmp
mpppp
mp
mnnn
mnnn
mnnn
n
n
mnnn
m
pn
m
pn
+
+
=
+++
+
=
+++
+++
−+++=−+++
==
( )
( )
( )
( ) ( ) ( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
2/,,3,2,1
!!!22
!221
!!!2!2
!
2
!22
1
!!2!2
!!22
!
!
2
1
1
0
0
2
2
02 nm
snssn
sn
a
snssnn
n
sn
a
mpmp
pmp
mp
p
a n
sthatsucha
choose
s
nmps
m
m =
−−
−−
=
−−
−
−=
+
+
−
−=
−
−
−=
( ) ( )
( ) ( )∑=
−−
<
−−
−
−=
2/
0
2
1
!2!!2
!22
1
n
s
sn
n
s
even xx
snsns
sn
y 36
Series Solutions – Frobenius’ Method
Legendre Polynomials
37. SOLO
Example: Legendre Equation (continue – 8)
3. The recurrence relation for odd x powers results in the following expression for the
coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) 2
1
,,3,2,1
!12
24213212
1 02
−
=
+
+++−+−+−
−=
n
ma
m
mnnnnmnmn
a
m
m
( ) ( ) ( ) ( )( ) ( )
( )!
!
222242222213212 1
12
mp
p
ppmpmpnmnmn m
pn
−
=−+−+−=−+−+− −
+=
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( )
( )
( ) ( )!!122
!!122
212
224222
1225232
!12
!12
242
1
1
12
mpp
pmp
mppp
mppp
mppp
p
p
mnnn
m
m
pn
++
++
=
+++
+++
++++
+
+
=+++
−
−
+=
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( )
( ) ( ) 2
1
,,3,2,1
!2!1!!
!
2
1
!122
1
!12!!!12
!!122
1 0
2
0
2
2
−
=
−−−
−
−−
−=
++−+
++
−=
−
−=
n
ma
snsnsn
n
sn
a
mmpmpp
pmp
a
sp
mps
m
m
37
( )
( ) ( )
( )[ ] ( ) ( )
( )
( )
( ) ( ) 2
1
,...,1,0
!2!!!2
!
2
1
!22
1
!2!12!!
!
2
1
!12222
1 0
2
0
2
−
=
−−
−
−
−=
−−−−
−
−−−
−=
−
−=
−
−=
n
sa
snsnsn
n
sn
a
snsnsnsn
n
snsn
sp
mps
sp
mps
Series Solutions – Frobenius’ Method
Legendre Polynomials
38. Example: Legendre Equation (continue – 9)
3. The recurrence relation for odd x powers results in the following expression for the
coefficients
( ) ( )
( ) ( )
( ) ( )
( ) ( ) 2
1
,...,1,0
!2!!2
!22
1
!2!!
!22
!
2
1
!2
1
1
0
0
2
2
1
2
−
=
−−
−
−=
−−
−
−
−=
−−
− n
s
snsns
sn
a
snsns
snn
n
a n
s
thatsucha
Choose
s
n
m
( ) ( ) ( )
( ) ( )
( )
1&
!2!!
!22
1
2
1 2/1
0
2
<
−−
−
−= ∑
−
=
−−
xoddnx
snsns
sn
xy
n
s
sns
nodd
We also found that the solution for k = 0 and a1 = 0 is
( ) ( ) ( )
( ) ( )
1&
!2!!2
!22
1
2/
0
2
<
−−
−
−= ∑=
−−
xevennx
snsns
sn
xy
n
s
sn
n
s
even
We recover the result obtained by the Generating Function of Legendre Polynomials
and by Rodrigues’ Formula 38
Therefore we can unify those two relations to obtain:
( ) ( ) ( )
( ) ( )
[ ]
1
!2!!2
!22
1
2/
0
2
<
−−
−
−= ∑=
−−
xx
snsns
sn
xP
n
s
sn
n
s
n
Series Solutions – Frobenius’ Method
SOLO
Legendre Polynomials
39. SOLO
Example: Legendre Equation (continue – 10)
2. Using k = 0 and a1 ≠ 0 we obtain an infinite series and not a polynomial.
This is the Second Solution of the Legendre Differential Equation.
The solution is the sum of the two infinite series, one with even powers of x
and the other with odd powers of x. The series solution, in this case,
diverges
at x = ± 1.
Those are Legendre Functions of the Second Kind.
The Polynomial solutions are Legendre Functions of the First Kind.
39
( ) ( )
( ) ( )
( ) ( )
( ) ( )
,2,1,0
21
1
21
11
2 =
++
++−
−=
++
+−+
−=+ j
jj
jnjn
a
jj
jjnn
aa jjj
In this case we have
The recurrence relation results in the following expression for the coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
,3,2,1
!2
123124222
1 02 =
−+++⋅−+−+−
−= ma
m
mnnnnnmnmn
a
m
m
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
,3,2,1
!12
123124222
1 112 =
+
−+++⋅−+−+−
−=+ ma
m
mnnnnnmnmn
a
m
m
Series Solutions – Frobenius’ Method
Legendre Polynomials
40. SOLO
Example: Legendre Equation (continue – 11)
40
We want to find a series that converges for |x| > 1.
Let return to the conditions to have a series solution for Legendre ODE
For k = 0 and a0 = 0
By substituting j = m – 2 we obtain
( ) ( )
( ) ( )
,2,1,0
1
21
2 ==
++−
++
−= + ja
jnjn
jj
a jj
( )
( ) ( ) mm a
mnmn
mm
a
12
1
2
−++−
−
−=−
( ) ( ) .0121 2
2
22
constrealryrr
xd
yd
yx
xd
yd
x =++−−
( ) ( )
( ) ( )
( ) ( )
( ) ( )
,2,1,0
21
1
21
11
2 =
++
++−
−=
++
+−+
−=+ j
jj
jrjr
a
jj
jjrr
aa jjj
We can make am+2, am+4, am+6,…. To vanish for m = r = n or m = -n – 1.
If we start for am ≠ 0 we can obtain the following recursive formula
Series Solutions – Frobenius’ Method
Legendre Polynomials
41. SOLO
Example: Legendre Equation (continue – 12)
41
( )
( ) ( ) mm a
mnmn
mm
a
12
1
2
−++−
−
−=−
Tacking m = n we obtain
( )
( )
( ) ( )
( )
( )( ) ( )
( ) ( ) nnn
nn
a
nn
nnnn
a
n
nn
a
a
n
nn
a
321242
321
324
32
122
1
24
2
−⋅−⋅⋅
−−−
+=
−
−−
−=
−
−
−=
−−
−
The first solution can be written as
( ) ( )
( )
( )( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
+
+−−⋅−⋅⋅
−−−−
++
−⋅−⋅⋅
−−−
+
−
−
−= −−−
innnn
n x
innni
ininnn
x
nn
nnnn
x
n
nn
xaxy 242
1
1223212242
1221
321242
321
122
1
Using d’Alembert – Cauchy test for convergence of an Infinite Series, we have
( )( )
( )
≤≥
><
=
+−
−−−
= −−
∞→+−
+−
−
−
∞→ divergex
convergex
xx
ini
inin
xa
xa
iin
in
in
in
j 11
11
1222
122
limlim 22
22
22
2
2
Series Solutions – Frobenius’ Method
Legendre Polynomials
42. SOLO
Example: Legendre Equation (continue – 13)
42
( )
( ) ( ) mm a
mnmn
mm
a
12
1
2
−++−
−
−=−
Tacking m =- n - 1 we obtain
( ) ( )
( )
( ) ( )
( )
( ) ( )( ) ( )
( ) ( ) 135
13
523242
4321
524
43
322
21
−−−−−−
−−−−
+⋅+⋅⋅
++++
+=
+
++
=
+
++
+=
nnn
nn
a
nn
nnnn
a
n
nn
a
a
n
nn
a
The second solution can be written as
( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
+
+++⋅+⋅⋅
+−+++
++
+⋅+⋅⋅
++++
+
+
++
+= −−−−−−−−−
−−
12531
12
1225232242
21221
523242
4321
322
21 innnn
n x
innni
ininnn
x
nn
nnnn
x
n
nn
xaxy
Using d’Alembert – Cauchy test for convergence of an Infinite Series, we have
( ) ( )
( )
≤≥
><
=
++
+−+
= −−
∞→+−
+−
−−
−−−
∞→ divergex
convergex
xx
ini
inin
xa
xa
iin
in
in
in
j 11
11
1222
212
limlim 22
12
12
12
12
Series Solutions – Frobenius’ Method
Legendre Polynomials
43. SOLO
Example: Legendre Equation (continue – 14)
43
( )( )
!
1353212
n
nn
an
⋅⋅−−
=
By choosing
we have
( ) ( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1]
1223212242
1221
1
321242
321
122
1
[
!
1353212
2
42
1
>+
+−−⋅−⋅⋅
−−−−
−+
+
−⋅−⋅⋅
−−−
+
−
−
−
⋅⋅−−
==
−
−−
xx
innni
ininnn
x
nn
nnnn
x
n
nn
x
n
nn
xyxP
ini
nnn
n
Return to Table of Content
Finally ( ) ( ) ( )
( ) ( )
1
!2!!2
!22
1
0
2
>
−−
−
−= ∑
∞
=
−
xx
inini
in
xP
i
in
n
i
n
( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )!2!!2
!22
1
!2
!22
!2!2
1
1
!2!2
135122
1
1223212242
1221
1
!
1353212
inini
in
in
in
iiniin
in
innni
ininnn
n
nn
n
i
ini
i
i
i
i
−−
−
−=
−
−
−
−=
−
⋅⋅−−
−=
+−−⋅−⋅⋅
−−−−
−
⋅⋅−−
−
Series Solutions – Frobenius’ Method
Legendre Polynomials
44. SOLO
Example: Legendre Equation (continue – 14)
44
( )( ) 1351212
!
1
⋅⋅−+
== −−
nn
n
aa nn
By choosing
we have
( ) ( )
( )( )
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
1]
1225232242
21221
523242
4321
322
21
[
1351212
!
12
531
2
>+
+++⋅+⋅⋅
+−+++
+
+
+⋅+⋅⋅
++++
+
+
++
+
⋅⋅−+
==
−−−
−−−−−−
xx
innni
ininnn
x
nn
nnnn
x
n
nn
x
nn
n
xyxQ
in
nnn
n
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )!122!
!2!
!
!12
!
!2
!122
!12
!
!2
2
1
!122
2222!12
!
!2
!2
1
1225232242
21221
2
++
+++
=
+
++
++
=
++
++++
=
+++⋅+⋅⋅
+−+++
ini
inin
n
n
n
in
in
n
n
in
in
innn
n
in
iinnni
ininnn
i
i
i
( ) ( ) ( )
( )
( )
1
!122!
!2!
2
0
12
>
++
++
= ∑
∞
=
++−
xx
ini
inin
xQ
i
inn
n
Go to Neumann Integral
( )( ) ( )!12
!2
!
1351212
!
+
=
⋅⋅−+ n
n
n
nn
n n
Finally
Series Solutions – Frobenius’ Method
Legendre Polynomials
45. SOLO
Example: Legendre Equation (continue – 15)
45
Summarize
( ) ( ) ( )
( )
( )
1
!122!
!2!
2
0
12
>
++
++
= ∑
∞
=
++−
xx
ini
inin
xQ
i
inn
n
Return to Table of Content
( ) ( ) ( )
( ) ( )
1
!2!!
!22
1
2
1
0
2
>
−−
−
−= ∑
∞
=
−
xx
inini
in
xP
i
ini
nn
( ) ( ) ( )
( ) ( )
[ ]
1
!2!!
!22
1
2
1 2/
0
2
<
−−
−
−= ∑=
−−
xx
inini
in
xP
n
i
ini
nn
Return to
Similar to Rodrigues Formula
Using Frobenius’ Method we found that solutions of Legendre ODE
are
( ) ( ) integerpositive0121 2
2
22
nynn
xd
yd
yx
xd
yd
x =++−−
Series Solutions – Frobenius’ Method
Legendre Polynomials
( )
4,3,2,1
1
=
>
l
xxPl
( )
4,3,2,1
1
=
>
l
xxQl
46. 46
SOLO
( ) 1
21
1
0
2
≤=
+−
∑
∞
=
utPu
utu n
n
n
For u=0 we obtain ( ) 10 =tP
For t = 1 we obtain ( ) ( ) 111
1
1
21
1
00
2
=⇒==
−
=
+−
∑∑
∞
=
∞
=
n
n
n
n
n
n
PPuu
uuu
For t = -1 we obtain ( ) ( ) ( ) ( )n
n
n
n
n
n
nn
PPuu
uuu
1111
1
1
21
1
00
2
−=⇒=−=
+
=
++
∑∑
∞
=
∞
=
Let find a Recursive Relation for Legendre Polynomial computation
Start with:
( ) ( )
( ) ( )
( )
( )
( )
∂
∂
+
−
=
∂
∂−
+
−
=
∂
∂
+
−
= ++
+
+
++
+
+
11
1
1
11
2
1 cos
1
11
!
1
1
11
!1
1cos
n
n
n
nn
n
nn
n
n
r
P
znrznnrznr
P θθ
( )
( )
( ) ( ) ( )( ) ( )
∂
∂
+
∂
∂+
−
+
−
= +++
+
zd
Pd
rz
r
r
Pn
nr
P n
nn
n
n
n θ
θ
θθθ cos
cos
cos1cos1
1
1cos
122
1
Recursive Relations for Legendre Polynomial Computation
First Recursive Relation
Legendre Polynomials
47. 47
SOLO
Recursive Relations for Legendre Polynomial Computation
( )
( )
( ) ( ) ( )( ) ( )
∂
∂
+
∂
∂+
−
+
−
= +++
+
zd
Pd
rz
r
r
Pn
nr
P n
nn
n
n
n θ
θ
θθθ cos
cos
cos1cos1
1
1cos
122
1
θcosrz =
θcos=
∂
∂
z
r
z
r
z
r
z
z
∂
∂
+
∂
∂
=
∂
∂
=
θ
θ
θ
cos
cos1
cos
rz
θθ 2
cos1cos −
=
∂
∂
( )
( )
( ) ( ) ( )( )
−
+
+
−
+
−
= +++
+
rd
Pd
rr
Pn
nr
P n
nn
n
n
n θ
θ
θ
θ
θθ 2
122
1 cos1
cos
cos1
cos
cos1
1
1cos
( ) ( ) ( )( )
θ
θθ
θθθ
cos
cos
1
cos1
coscoscos
2
1
d
Pd
n
PP n
nn
+
−
−=+
Substituting t = cos θ we obtain
( ) ( ) ( )
td
tPd
n
t
tPttP n
nn
1
1 2
1
+
−
−=+
First Recursive Relation (continue – 1)
Legendre Polynomials
48. 48
SOLO
Recursive Relations for Legendre Polynomial Computation
( ) ( ) ( ) 0,1
1
1 2
1 ≥≤
+
−
−=+ nt
td
tPd
n
t
tPttP n
nn
Use to start ( ) ( ) 01 0
0 =⇒=
td
tPd
tP
( )
( )
( )
( )
( )
8
157063
8
33035
2
35
2
13
35
5
24
4
3
3
2
2
1
ttt
tP
tt
tP
tt
tP
t
tP
ttP
+−
=
+−
=
−
=
−
=
=
First Recursive Relation (continue – 2)
Legendre Polynomials
49. 49
SOLO
Recursive Relations for Legendre Polynomial Computation
Start from ( ) ( ) 1
21
1
:,
0
2
≤=
+−
= ∑
∞
=
utPu
utu
tug
n
n
n
Let differentiate both sides with respect to u and rearranging
( ) ( )∑
∞
=
−
+−=
+−
−
=
∂
∂
1
12
2
21
21 n
n
n
tPunutu
utu
ut
u
g
( ) ( ) ( ) ( )∑∑
∞
=
−
∞
=
+−=−
1
12
0
21
n
n
n
n
n
n
tPunututPuut
( ) ( ) ( ) ( )∑∑
∞
=
+−
∞
=
+
+−=−
1
11
0
1
2
n
n
nnn
n
n
nn
tPunutnuntPuut
( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑∑∑∑
∞
=
−
∞
=
∞
=
+
∞
=
−
∞
=
−+−+=−
2
1
10
1
1
1
0
121
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
tPuntPutntPuntPutPut
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
≥+=−+
=−=−
==
+− 2112
122
0
11
1201
10
ntPuntPuntPutn
ntPtutPutPutuPt
ntPtPt
n
n
n
n
n
n
We can see that the last relation agrees also with the previous
relations, for n = 0 and n = 1.
Second Recursive Relation
Legendre Polynomials
51. 51
SOLO
Recursive Relations for Legendre Polynomial Computation
Start from ( ) ( ) 1
21
1
:,
0
2
≤=
+−
= ∑
∞
=
utPu
utu
tug
n
n
n
Let differentiate both sides with respect to t and rearranging
( )
( )
∑
∞
=
=
+−
=
∂
∂
0
2/32
21 n
nn
td
tPd
u
utu
u
t
g
( ) ( ) ( )
∑∑
∞
=
∞
=
+−=
0
2
0
21
n
nn
n
n
n
td
tPd
uututPuuor
Equaling coefficients of each power of u gives
( ) ( ) ( ) ( )
td
tPd
td
tPd
t
td
tPd
tP nnn
n
11
2 −+
+−=
( ) ( ) ( ) ( ) ( )tPntPntPtn nnn 11112 −+ ++=+
Differentiate the Second Recursive Relation
with respect to t and rearranging
( ) ( ) ( )
( )
( )
( )
( )
td
tPd
n
n
td
tPd
n
n
td
tPd
ttP nnn
n
11
1212
1 −+
+
+
+
+
=+
Third Recursive Relation
Legendre Polynomials
52. 52
SOLO
Recursive Relations for Legendre Polynomial Computation
We found ( ) ( ) ( ) ( )
td
tPd
ttP
td
tPd
td
tPd n
n
nn
211
+=+ −+
( )
( )
( )
( )
( ) ( ) ( )
td
tPd
ttP
td
tPd
n
n
td
tPd
n
n n
n
nn
+=
+
+
+
+ −+ 11
1212
1
Third Recursive Relation (continue)
Let solve for and in terms of and( )tPn
( )
td
tPd n( )
td
tPd n 1+( )
td
tPd n 1−
( ) ( ) ( )
td
tPd
ttPn
td
tPd n
n
n
+−=−1
( ) ( ) ( ) ( )
td
tPd
ttPn
td
tPd n
n
n
++=+
11
Subtracting the first relation from the second gives the Third Recursive Relation
( ) ( ) ( ) ( )
td
tPd
td
tPd
tPn nn
n
11
12 −+
−=+
Legendre Polynomials
53. 53
SOLO
Recursive Relations for Legendre Polynomial Computation
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }yPxPyPxP
yx
n
yPxPk
tPtntPn
td
tPd
t
tPntPtn
td
tPd
t
tPn
td
tPd
t
td
tPd
tPn
td
tPd
td
tPd
t
tPn
td
tPd
td
tPd
tPntPtntPn
tP
n
n
tP
n
n
tPt
tPin
td
tPd
nnnn
n
k
kk
nn
n
nn
n
n
nn
n
nn
n
nn
nnn
nnn
l
i
in
n
11
0
1
2
1
2
1
1
1
11
11
11
1
2
1
0
12
1
129
1118
17
6
5
124
01213
1212
1
2
1421
++
=
+
−
−
−
−
−+
−+
−+
−
=
−−
−
−
+
=+
+−+=−
−=−
=−
=−
+=−
=++−+
+
+
+
+
=
−−=
∑
∑
Recursive Relation between Legendre Polynomials and their Derivatives)
Legendre Polynomials
Return to Table of Content
54. 54
SOLO
Orthogonality of Legendre Polynomials
Define ( ) ( )tPwtPv nm == :&:
We use Legendre’s Differential Equations:
( ) ( ) 011 2
=++
− vmm
td
vd
t
td
d
( ) ( ) 011 2
=++
− wnn
td
wd
t
td
d
Multiply first equation by w and integrate from t = -1 to t = +1.
( ) ( ) 011
1
1
1
1
2
=++
− ∫∫
+
−
+
−
dtwvmmdtw
td
vd
t
td
d
Integrate the first integral by parts we get
( ) ( ) ( ) 0111
1
1
1
1
2
0
1
1
2
=++−−− ∫∫
+
−
+
−
+=
−=
dtwvmmdt
td
wd
td
vd
tw
td
vd
t
t
t
In the same way, multiply second equation by v and integrate from t = -1 to t = +1.
( ) ( ) 011
1
1
1
1
2
=++−− ∫∫
+
−
+
−
dtwvnndt
td
wd
td
vd
t
Legendre Polynomials
55. 55
SOLO
Orthogonality of Legendre Polynomials
( ) ( ) 011
1
1
1
1
2
=++−− ∫∫
+
−
+
−
dtwvmmdt
td
wd
td
vd
t
Subtracting those two equations we obtain
( ) ( ) 011
1
1
1
1
2
=++−− ∫∫
+
−
+
−
dtwvnndt
td
wd
td
vd
t
( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 01111
1
1
1
1
=+−+=+−+ ∫∫
+
−
+
−
dttPtPnnmmdtwvnnmm nm
This gives the Orthogonality Condition for m ≠ n
( ) ( ) nmdttPtP nm ≠=∫
+
−
0
1
1
To find let square the relation and integrate
between t = -1 to t = +1. Due to orthogonality only the integrals of terms
having Pn
2
(t) survive on the right-hand side. So we get
( )∫
+
−
1
1
2
dttPn
( )∑
∞
=
=
+− 0
2
21
1
n
n
n
tPu
utu
( )∑ ∫∫
∞
=
+
−
+
−
=
+− 0
1
1
22
1
1 2
21
1
n
n
n
dttPudt
utu
Legendre Polynomials
56. 56
SOLO
Orthogonality of Legendre Polynomials
( )∑ ∫∫
∞
=
+
−
+
−
=
+− 0
1
1
22
1
1 2
21
1
n
n
n
dttPudt
utu
( ) ( )
( )
1
1
1
ln
1
1
1
ln
2
1
21ln
2
1
21
1
2
21
1
2
1
1 2
<
−
+
=
+
−
−
=−+
−
=
−+
+=
−=
+
−∫ u
u
u
uu
u
u
tuu
u
dt
tuu
t
t
( ) ( ) ( ) ( ) ( ) ( ) ( )
∑∑∑
∞ ++∞ +∞ +
+
−−
−=
+
−
−−
+
−=−−+
0
11
0
1
0
1
1
1
1
1
1
1
1
1
1
1ln
1
1ln
1
n
uu
un
u
un
u
u
u
u
u
u
nn
n
n
n
n
n
( ) ( ) ( )
( )
( ) ( )
( ) ∑∑∑∑
∞∞ +∞ ++
+
∞ ++
+
=
+
=
+
−−
−+
+
−−
−=
0
2
0
12
0
0
1212
12
0
1212
2
12
2
12
1
2
12
1
1
12
1
1 n
nnn
n
nn
n
u
nn
u
un
uu
un
uu
u
Let compute first
Therefore
( )∑ ∫∑∫
∞ +
−
∞+
−
=
+
=
+− 0
1
1
22
0
2
1
1 2
12
2
21
1
dttPuu
n
dt
utu
n
nn
Comparing the coefficients of u2n
we get ( )
12
21
1
2
+
=∫
+
− n
dttPn
Legendre Polynomials
( ) ( ) nmmn
n
dttPtP δ
12
21
1 +
=∫
+
−
Hence
57. 57
SOLO
Using Rodrigues’ Formula
let calculate
( ) ( ) ( ) ( )[ ]∫∫
+
−
+
−
−
=
1
1
2
1
1
1
!2
1
dt
td
td
tP
n
dttPtP n
nn
knnk
Legendre Polynomials
( ) ( )[ ]n
nn
nn
td
td
n
tP
1
!2
1 2
−
=
Orthogonality of Legendre Polynomials (Second Method)
Assume, without loss of generality, that n > k, and integrate by parts
( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )
∫∫
∫∫
+
−
+
− −
−
+
−
−
−
+
−
+
−
−−==
−
−
−
=
−
=
1
1
2
1
1 1
21
0
1
1
1
21
1
1
2
1
1
1
!2
1
1
1
!2
11
!2
1
1
!2
1
dt
td
tPd
t
n
dt
td
td
td
tPd
ntd
td
tP
n
dt
td
td
tP
n
dttPtP
n
k
n
n
n
n
n
nn
k
nn
nn
kn
n
nn
knnk
Since Pk (t) is a Polynomial of Order k and we assume that n > k, we have
( ) 0=n
k
n
td
tPd
Therefore
( ) ( ) nkdttPtP nk ≠=∫
+
−
0
1
1
58. 58
SOLO
For k = n we have
Legendre Polynomials
Orthogonality of Legendre Polynomials (Second Method) (continue – 1)
( )[ ] ( ) ( ) ( )
∫∫
+
−
+
−
−−=
1
1
2
1
1
2
1
!2
1
1 dt
td
tPd
t
n
dttP n
n
n
n
n
n
n
But we found that Pn (t) is given by:
( ) ( ) ( )
( ) ( )
[ ]
∑=
−
−−
−
−=
2/
0
2
!2!!2
!22
1
n
k
kn
n
k
n t
knknk
kn
tP
Therefore to compute it is sufficient to consider only the highest power of t
in the series, i.e. for k = 0, and we obtain
( )
n
n
n
td
tPd
( ) ( )
( )
( )
!2
!2
!
!2
!2
2
n
n
n
n
n
td
tPd
nnn
n
n
==
( )[ ] ( ) ( ) ( ) ( )
( )
( )∫∫∫
+
−
+
−
+
−
−=−−=
1
1
2
22
1
1
2
1
1
2
1
!2
!2
!2
!2
1
!2
1
1 dtt
n
n
dt
n
n
t
n
dttP
n
nn
n
n
n
n
59. SOLO
Legendre Polynomials
Orthogonality of Legendre Polynomials (Second Method) (continue – 2)
( )[ ] ( ) ( ) ( ) ( )
( )
( )∫∫∫
+
−
+
−
+
−
−=−−=
1
1
2
22
1
1
2
1
1
2
1
!2
!2
!2
!2
1
!2
1
1 dtt
n
n
dt
n
n
t
n
dttP
n
nn
n
n
n
n
( ) ∫∫∫
++
=+
−
=−=−
π
π
θ
θθθθ
0
12
0
12
cos1
1
2
sinsin1 dddtt nn
tn
Therefore
( )
( ) ( ) ( ) ( ) ( )
( )
( )!12
!2
cos
2222
!2
11212
!2
sin
31212
2222
sin
12
2
sin
212
2
0
1
00
12
0
12
+
=
−⋅
⋅
−⋅+
=
−⋅+
−⋅
=
+
=
+
−+
∫∫∫ n
n
nn
n
nn
n
d
nn
nn
d
n
n
d
nnn
nn
π
πππ
θθθθθθθ
( ) ( ) ( )∫∫∫∫
+−−+
−=+−==
ππ
π
ππ
θθθθθθθθθθθθ
0
1212
0
212
0
0
2
0
2
0
12
sinsin2cossin2cossincossinsin dndndd nnnnnn
( )[ ] ( )
( )
( ) ( )
( )
( )
( )
,2,1,0
12
2
!12
!2
!2
!2
1
!2
!2
212
22
1
1
2
22
1
1
2
=
+
=
+
=−=
+
+
−
+
− ∫∫ n
nn
n
n
n
dtt
n
n
dttP
n
n
n
nn
Return to Table of Content
59
60. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Using Sturm-Liouville Theory it can be seen that the Legendre
Polynomials that are Solution of the Legendre ODE, form an orthogonal
and “Complete” Set, meaning that we can expand any function f (t) ,
Piecewise Continuous in the interval -1 ≤ t ≤+1. Therefore we can define a
series of Legendre Polynomials that converges in the mean to the function
f (t)
( ) ( ) 11
0
≤≤−= ∑
∞
=
ttPatf
n
nn
The coefficients an can be defined using the
Orthogonality Property of Legendre Polynomials
( ) ( ) ( ) ( ) m
n
m
mnnm a
m
tdtPtPatdtPtf
mn
12
2
0
2
12
1
1
1
1
+
== ∑ ∫∫
∞
=
+
+
−
+
−
δ
( ) ( )∫
+
−
+
=
1
1
2
12
tdtPtf
m
a mm
( ) ( ) ( ) ( ) 11
2
12
0
1
1
≤≤−
+
= ∑ ∫
∞
=
+
−
ttPtdtPtf
n
tf
n
nn
60
61. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
At any discontinuous point t0 ( f(t0- ) ≠ f(t0+) ) we have
( ) ( )[ ] ( ) ( ) ( ) 11
2
12
2
1
0
0
0
1
1
00 ≤≤−
+
=+ ∑ ∫
∞
=
+
−
+− ttPtdtPtf
n
tftf
n
nn
If f (t) is defined in the interval –a ≤ t ≤ +a then
( ) ( ) ataatPatf
n
nn ≤≤−= ∑
∞
=0
/ ( ) ( )∫
+
−
+
=
a
a
nn tdatPtf
a
n
a /
2
12
61
62. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
If f (t) is an odd function ( f(-t ) =- f(t) ) we have
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
=+−−=
+==
+
∫
∫∫
∫∫∫
+
++
−−
+
−→
−
+
−
1
0
1
0
1
0
1
1
0
0
1
1
1
2
0
12
2
oddndttPtf
evenn
dttPtfdttPtf
dttPtfdttPtfdttPtfa
n
n
n
tP
n
tf
n
tt
nnn
n
n
If f (t) is an even function ( f(-t ) = f(t) ) we have
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
=+−−=
+==
+
∫∫∫
∫∫∫
+
++
−
+
−→
−
+
−
oddn
evenndttPtf
dttPtfdttPtf
dttPtfdttPtfdttPtfa
n
n
n
tP
n
tf
n
tt
nnn
n
n 0
2
12
2
1
0
1
0
1
0
1
1
0
0
1
1
1
62
63. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Using Rodrigues’ Formula
let calculate
( ) ( )[ ]n
nn
nn
td
td
n
tP
1
!2
1 2
−
=
( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )
∫∫
∫∫
+
−
+
−
−
−
+
−
−
−
+
−
+
−
−−==
−
−
−
=
−
=
1
1
2
1
1
1
21
0
1
1
1
21
1
1
21
1
1
!2
1
1
1
!2
11
!2
1
1
!2
1
td
td
tfd
t
n
td
td
tfd
td
td
ntd
td
tf
n
tdtf
td
td
n
tdtftP
n
n
n
n
n
n
nn
nn
nn
n
n
nn
nn
( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫∫
+
−
+
−
+
−
−=−−=
1
1
2
1
1
2
1
1
1
!2
1
1
!2
1
1 td
td
tfd
t
n
td
td
tfd
t
n
tdtPtf n
n
n
nn
n
n
n
n
n
or
63
64. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Example f (t) = tk
,|t| < 1
( ) ( ) ( ) ( ) ( ) ( ) ( )
≥−+−−
+
<
=−
+
=
+
=
∫
∫∫ +
−
−
+
+
−
+
+
−
nktdttnkkk
n
n
nk
td
td
td
t
n
n
tdtPt
n
a nkn
n
n
kn
n
nn
k
n
1
1
2
1
1
1
2
1
1
1
111
!2
12
0
1
!2
12
2
12
We have
( )
( ) ( )
( ) ( )
( )
( )( )
( ) ( )
( )
( )( ) ( )[ ]
( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )∫
∫
∫
∫∫
+
−
+
−−
+
−
−−−+
+
−
−−+
+
−
−−+
+
−
−−+=
−=
+
−
−
−
−
−
+
=
−
+++
−−−−−−−−
=
=−
++
−−−−
=
−
+
−−
+−
+
−=−
−−
1
1
2
1
1
22222
1
1
422222
1
1
222122
0
1
1
122122
1
2
1
1
1
2222
1
!2
!22
!2
!2
1
2222122
12222322122
1
222122
322122
1
122
122
1
122
1
1
122
222
tdt
nm
nm
mn
n
tdtt
mnnn
nmnmnmnm
tdtt
nn
nmnm
tdtt
n
nm
tt
n
tdtt
mn
nmnm
nmnmmn
nmn
nmnnmntu
tdtdv
nmn
nm
n
Take k = 2m and since t2m
is even, they are only even coefficients nonzero so
we take 2 n instead of n, and 2n ≤ 2m
64
65. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Example f (t) = tk
, |t| < 1 (continue – 1)
( )
( )
( )
( )
( )
≥−
−
+
<
=
∫
+
−
−
+
nmtdtt
nm
m
n
n
nm
a nmn
n
n
1
1
2222
12
2
1
!22
!2
!22
14
0
We have for f (t) = t2m
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )( )
( )!122
!2
!2
!22
!2
!2
1
!2
!22
!2
!2
1
2122
1
1
2
1
1
2222
++
+
−
−
+
=
−
−
−
+
=−
++
−−
+
−
+
−−
+
−
−
∫∫
nm
mn
nm
nm
mn
n
tdt
nm
nm
mn
n
tdtt
nm
nmnm
mn
nmnm
nmn
( )
( )
( )
( )
( )
( )
( )
( )
( )( )
( )
( ) ( ) ( )
( ) ( )!122!
!!2142
!122
!2
!2
!22
!2
!2
!22
!2
!22
14 22122
122
++−
++
=
++
+
−
−
+−
+
=
++
−−+
nmnm
nmmn
nm
nm
nm
nm
nm
n
nm
m
n
n
a
nnm
nmnmnn
( ) ( )
( )!12
!2
1
212
1
1
2
+
=−
+
+
−∫ n
n
dtt
n
n
Where we used the previous result
Therefore
( ) ( ) ( )
( ) ( )
( )∑= ++−
++
=
m
n
n
n
m
tP
nmnm
nmmn
t
0
2
2
2
!122!
!!2142
65
66. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
( ) ( ) ( ) ( ) ( ) ( ) ( )
≥−+−−
+
<
=−
+
=
+
=
∫
∫∫ +
−
−
+
+
−
+
+
−
nktdttnkkk
n
n
nk
td
td
td
t
n
n
tdtPt
n
a nkn
n
n
kn
n
nn
k
n
1
1
2
1
1
1
2
1
1
1
111
!2
12
0
1
!2
12
2
12
We have
( )
( ) ( )
( ) ( )
( )
( )( )
( ) ( )
( )
( )( ) ( )[ ]
( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )( )
( )
( )∫
∫
∫
∫∫
+
−
++
++
−−
+
−
−−−++
+
−
−−+
+
−
−−+
+
−
−−+=
−=
+
−
−+
−
++
+
−
−
++
+
=
−
++++
−−−−−−−−
=
=−
++
−−−−
=
−
+
−−
+−
+
−=−
−−
+
1
1
12
2122
1
1
222212
1
1
422322
1
1
222222
0
1
1
122222
1
2
1
1
1
22122
1
!122
!2
!2
!22
!12
!12
1
12322222
12222322122
1
322222
322122
1
222
122
1
222
1
1
122
2122
tdt
nm
mn
nm
nm
mn
n
tdtt
mnnn
nmnmnmnm
tdtt
nn
nmnm
tdtt
n
nm
tt
n
tdtt
mn
nm
nmnm
nmnmmn
nmn
nmnnmntu
tdtdv
nmn
nm
n
Take k = 2m+1 and since t2m+1
is odd, they are only odd coefficients nonzero so
we take 2 n+1 instead of n, and 2n+1 ≤ 2m+1
66
Example f (t) = tk
, |t| < 1 (continue – 2)
67. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
( )
( )
( )
( )
( )
≥−
−+
+
<
=
∫
+
−
−+
+
+
nmtdtt
nm
m
n
n
nm
a nmn
n
n
1
1
22122
22
12
1
!22
!2
!122
34
0
We have for f (t) = t2m+1
( ) ( )
( )
( )
( )
( )( )
( )
( )
( )
( )
( )
( )
( )( )
( )!322
!12
!2
!22
!12
!12
1
!122
!2
!2
!22
!12
!12
1
2322
1
1
12
21221
1
22122
++
++
−
−
++
+
=
−
++
+
−
−
++
+
=−
++
−−
+
−
++
++
−−
+
−
−+
∫∫
nm
mn
nm
nm
mn
n
tdt
nm
mn
nm
nm
mn
n
tdtt
nm
nmnm
mn
nm
nmnm
nmn
( )
( )
( )
( )
( )
( )
( )
( )
( )( )
( )
( ) ( ) ( )
( ) ( )!322!
!1!12342
!322
!12
!2
!22
!12
!12
!22
!2
!122
34 122322
2212
++−
++++
=
++
++
−
−
++
+
−+
+
=
+++
−−++
nmnm
nmmn
nm
nm
nm
nm
nm
n
nm
m
n
n
a
nnm
nmnmnn
( ) ( )
( )!12
!2
1
212
1
1
2
+
=−
+
+
−∫ n
n
dtt
n
n
Where we used the previous result
Therefore
( ) ( ) ( )
( ) ( )
( )∑=
+
+
+
++−
++++
=
m
n
n
n
m
tP
nmnm
nmmn
t
0
12
12
12
!322!
!1!12342
Return to Neumann Integral
67
Example f (t) = tk
, |t| < 1 (continue – 3)
68. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Neumann Integral
1
1
1
111
0
2
12
0
12
2
0
1
0
<+==
=
−
=
−
∑∑∑∑
∞
=
−∞
=
+
∞
=
+
∞
= x
t
x
t
x
t
x
t
x
t
x
x
txtx m
m
m
m
m
m
m
m
m
m
m
Start from
Use
( ) ( ) ( )
( ) ( )
( )∑=
+
+
+
++−
++++
=
m
n
n
n
m
tP
nmnm
nmmn
t
0
12
12
12
!322!
!1!12342
( ) ( ) ( )
( ) ( )
( )∑= ++−
++
=
m
n
n
n
m
tP
nmnm
nmmn
t
0
2
2
2
!122!
!!2142
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )
( )
( )∑ ∑∑ ∑
∑ ∑∑ ∑
∑ ∑∑ ∑
∞
=
∞
=
+
−
+∞
=
∞
=
++−
+=
∞
=
∞
=
+
−
+∞
=
∞
=
+−
∞
= =
−
+
+∞
=
+−
=
++
+++++
+
++
+++
=
++−
++++
+
++−
++
=
++−
++++
+
++−
++
=
−
0 0
12
2
12
0 0
2
212
2
0
12
2
12
0
2
12
2
0 0
2
12
12
0
12
0
2
2
!322!
!12!122342
!124!
!2!22142
!322!
!1!12342
!122!
!!2142
!322!
!1!12342
!122!
!!21421
n i
n
m
n
n i
n
in
ninm
n nm
n
m
n
n nm
n
m
nOrder
Summation
Change
m
m
n
m
n
n
m
m
m
n
n
n
tPx
nmi
ininn
tPx
ini
ininn
tPx
nmnm
nmmn
tPx
nmnm
nmmn
xtP
nmnm
nmmn
xtP
nmnm
nmmn
tx
68
69. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Neumann Integral (continue – 1)
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )
( )
( )∑ ∑∑ ∑
∞
=
+
∞
=
−
+∞
=
∞
=
++−
++
+++++
+
++
+++
=
− 0
12
0
2
12
0
2
0
212
2
!322!
!12!122342
!124!
!2!221421
n
n
i
m
n
n
n
i
in
n
tPx
nmi
ininn
tPx
ini
ininn
tx
We found
( ) ( ) ( )
( )
( )
1
!122!
!2!
2
0
12
>
++
++
= ∑
∞
=
++−
xx
ini
inin
xQ
i
inn
n
Use the Frobenius Series development of Legendre Functions of the
Second Kind Qn (x)
We have ( ) ( ) ( ) ( ) ( ) ( )∑∑
∞
=
++
∞
=
+++=
− 0
1212
0
22 3414
1
n
nn
n
nn tPxQntPxQn
tx
( ) ( ) ( )∑
∞
=
<+=
− 0
12
1
n
nn xttPxQn
tx
Return to Qn
Frobenius Series
69
70. SOLO
Legendre Polynomials
Expansion of Functions, Legendre Series
Neumann Integral (continue – 2)
We found
( ) ( ) ( )∑
∞
=
+=
− 0
12
1
n
nn tPxQn
tx
Multiply both sides by Pm (t) and integrate between -1 to +1
( )
( ) ( ) ( ) ( ) ( )xQtdtPtPxQntd
tx
tP
m
n
n
mnn
m
nm
212
0
12
2
1
1
1
1
=+=
−
∑ ∫∫
∞
=
+
+
−
+
−
δ
We obtain
( )
( )
∫
+
−
−
=
1
1
2
1
td
tx
tP
xQ
n
n
Franz Neumann's Integral of 1848
Franz Ernst Neumann
(1798 –1895)
Return to Table of Content
70
71. SOLO
71
Legendre Polynomials
Schlaefli Integral
Start with
Using Rodrigues's Formula we obtain( ) ( )[ ]n
n
n
nn x
xd
d
n
xP 1
!2
1 2
−=
( ) ( )
∫ −
= td
zt
tf
j
zf
π2
1
Cauchy's Integral
with ( ) ( )n
zzf 12
−=
( ) ( )
∫ −
−
=− td
zt
t
j
z
n
n 1
2
1
1
2
2
π
Differentiate n times this equation with respect to z and multiply by 1/ (2n
n!)
( ) ( )
( )∫ +
−
−
−
=− td
zt
t
j
z
zd
d
n n
nn
n
n
n
n 1
2
2 1
2
2
1
!2
1
π
with the contour enclosing the point t = z.
Schlaefli Integral ( ) ( )
( )∫ +
−
−
−
= td
zt
t
j
zP n
nn
n 1
2
1
2
2
π
Return to Table of Content
73. SOLO
73
Legendre Polynomials
Laplace’s Integral Representation (continue -
1)
{ } ( )∑∑ ∫
∞
=
∞
=
=
+−
=−±
0
2
0 0
2
21
cos1
n
n
n
n
n
n
xPu
uxu
dxxu π
π
φφ
π
We obtained
Equating un
coefficients we obtain :
( ) { }∫ −±=
π
φφ
π 0
2
cos1
1
dxxxP
n
n
( ) ( ) 0112
=+−
− ynny
xd
d
x
xd
d
If we replace in the Legendre ODE n by –n – 1
the equation does not change. Therefore , and( ) ( )xPxP nn 1−−=
( ) { }∫
−−
−±=
π
φφ
π 0
1
2
cos1
1
dxxxP
n
n
Substitute x = cosθ
( ) { }∫ ±=
π
φφθθ
π
θ
0
cossincos
1
cos djP
n
n
Laplace’s First Integral
Laplace’s Second Integral
74. SOLO
74
Legendre Polynomials
Laplace’s Integral Representation (continue -
2)
Use the Generating Function
[ ] ( )∑
∞
=
=
+− 0
2/12
21
1
n
n
n
tPu
uut
Substitute t = cosθ and u = ejφ
[ ] ( )∑
∞
=
=
+− 0
2/12
cos
cos21
1
n
n
nj
jj
Pe
ee
θ
θ
ϕ
ϕϕ
[ ] [ ] [ ] [ ] [ ] 2/12/12/12/12/12
coscos2cos2cos21 θϕθθ ϕϕϕϕϕϕ
−=+−=+− − jjjjjj
eeeeeeBut
Therefore
( )
( )
( )
( )
>
−
<
−
=
−
∞
=
∑
θϕ
θϕ
θϕ
θϕ
θ
πϕ
ϕ
ϕ
2/12/
2/12/
0
coscos2
1
coscos2
1
cos
j
j
n
n
nj
e
e
Pe
Equating the real and
imaginary parts, we obtain
( ) ( )
( )
( )
( )
( )
>
−
<
−
=∑
∞
=
θϕ
θϕ
ϕ
θϕ
θϕ
ϕ
θϕ
2/1
2/1
0
coscos2
2/sin2
coscos2
2/cos2
coscos
n
nPn
( ) ( )
( )
( )
( )
( )
>
−
<
−
−
=∑
∞
=
θϕ
θϕ
ϕ
θϕ
θϕ
ϕ
θϕ
2/1
2/1
0
coscos2
2/cos2
coscos2
2/sin2
cossin
n
nPn
75. SOLO
75
Legendre Polynomials
Laplace’s Integral Representation (continue -
3)
Let multiply first relation by cos (nφ) and the second by sin (nφ) and integrate
over φ on (0,π), we obtain two integrals
( ) ( )
( )
( )
( )
( )
>
−
<
−
=∑
∞
=
θϕ
θϕ
ϕ
θϕ
θϕ
ϕ
θϕ
2/1
2/1
0
coscos2
2/sin2
coscos2
2/cos2
coscos
i
iPi ( ) ( )
( )
( )
( )
( )
>
−
<
−
−
=∑
∞
=
θϕ
θϕ
ϕ
θϕ
θϕ
ϕ
θϕ
2/1
2/1
0
coscos2
2/cos2
coscos2
2/sin2
cossin
i
iPi
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( )
( )∫ ∫∑∫ −
+
−
==
∞
=
θ π
θ
δ
π
π
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
θ
π
θϕϕϕ
0
2/12/1
0
2
0 coscos2
cos2/sin2
coscos2
cos2/cos2
cos
2
coscoscos d
n
d
n
PPdni n
i
i
in
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( )
( )∫ ∫∑∫ −
+
−
−==
∞
=
θ π
θ
δ
π
π
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
θ
π
θϕϕϕ
0
2/12/1
0
2
0 coscos2
sin2/cos2
coscos2
sin2/sin2
cos
2
cossinsin d
n
d
n
PPdni n
i
i
in
( ) ( ) ( )
( )
( ) ( )
( )
−
+
−
= ∫ ∫
θ π
θ
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
π
θ
0
2/12/1
coscos
cos2/sin
coscos
cos2/cos2
cos d
n
d
n
Pn
( ) ( ) ( )
( )
( ) ( )
( )
−
+
−
−= ∫ ∫
θ π
θ
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
π
θ
0
2/12/1
coscos
sin2/cos
coscos
sin2/sin2
cos d
n
d
n
Pn
Dirichlet Integrals
Johann Peter Gustav
Lejeune Dirichlet
(1805 –1859)
76. SOLO
76
Legendre Polynomials
Add and subtract those two equations
( ) ( ) ( )
( )
( ) ( )
( )
−
+
−
= ∫ ∫
θ π
θ
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
π
θ
0
2/12/1
coscos
cos2/sin
coscos
cos2/cos2
cos d
n
d
n
Pn
( ) ( ) ( )
( )
( ) ( )
( )
−
+
−
−= ∫ ∫
θ π
θ
ϕ
θϕ
ϕϕ
ϕ
θϕ
ϕϕ
π
θ
0
2/12/1
coscos
sin2/cos
coscos
sin2/sin2
cos d
n
d
n
Pn
Dirichlet Integrals
( ) ( )[ ]
( )
( )[ ]
( )
−
+
+
−
+
= ∫ ∫
θ π
θ
ϕ
θϕ
ϕ
ϕ
θϕ
ϕ
π
θ
0
2/12/1
coscos
2/1sin
coscos
2/1cos
2
1
cos d
n
d
n
Pn
( )[ ]
( )
( )[ ]
( )∫ ∫ −
−
−
−
−
=
θ π
θ
ϕ
θϕ
ϕ
ϕ
θϕ
ϕ
0
2/12/1
coscos
2/1sin
coscos
2/1cos
0 d
n
d
n
Replace n by n + 1 in the last equation and substitute in the previous
( ) ( )[ ]
( )
( )[ ]
( )∫ ∫ −
+
=
−
+
=
θ π
θ
ϕ
θϕ
ϕ
π
ϕ
θϕ
ϕ
π
θ
0
2/12/1
coscos
2/1sin2
coscos
2/1cos2
cos d
n
d
n
Pn
Mehler Integrals
Gustav Ferdinand
Mehler
(1835 - 1895)
Return to Table of Content
Laplace’s Integral Representation (continue -
4)
77. SOLO
77
Legendre Polynomials
We found
Return to Table of Content
Integrals in terms of sin(iθ) and cos(iθ)
( ) ( ) mnnk
n
dttPtP δ
12
21
1 +
=∫
+
−
( ) ( ) mnnk
n
dPP δθθθθ
π
12
2
sincoscos
0 +
=∫
θcos=t
( ) ( ) ( )
( ) ( )
≥
++−
+
<
=
+
= −
+
−
∫ nm
nmnm
nmm
nm
a
n
tdtPt n
nn
m
!122!
!!22
0
14
2 12
2
1
1
2
2
( ) ( ) ( )
( ) ( )
≥
++−
+++
<
=
+
= +
+
+
−
+
+
∫ nm
nmnm
nmm
nm
a
n
tdtPt n
nn
m
!322!
!1!122
0
34
2 22
12
1
1
12
12
( ) ( ) ( )
( ) ( )
≥
++−
+
<
= −
∫ nm
nmnm
nmm
nm
dP n
n
m
!122!
!!22
0
sincoscos 12
0
2
2
π
θθθθ
( ) ( ) ( )
( ) ( )
≥
++−
+++
<
= +
+
+
∫ nm
nmnm
nmm
nm
dP n
n
m
!322!
!1!122
0
sincoscos 22
0
12
12
π
θθθθ
78. Ordinary Differential EquationsSOLO
Second Order Linear Ordinary Differential Equation (ODE)
78
Legendre Functions of the Second Kind Qn (x)
( ) [ ]
( ) ( )∑
∫
∞
=
−−
∞ −−
=
−
−
+−
−+=
0
2
12/12
0
1
2
1
cosh21
cosh1
n
n
n
n
n
txQ
x
xt
ttx
dxxxQ θθ
( ) ( )n
n
n
nn x
xd
d
n
xP 1
!2
1 2
−=
( ) ( ) 1<+= xxQBxPAy nn
( ) ( ) ( )xP
xd
d
xxP nm
m
mm
n
2/2
1−=
( ) ( ) ( )xQ
xd
d
xxQ nm
m
mm
n
2/2
1−=
( ) ( ) ( )xW
x
x
xPxQ nnn 1
1
1
ln
2
1
−−
−
+
=
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
3
2
2
5
2
3
1
1
1
ln
2
1
2
033
022
011
0
+−=
−=
−=
−
+
=
xxQxPxQ
xxQxPxQ
xQxPxQ
x
x
xQ
( ) ( ) ( )∑=
−−− =
n
m
mnmn xPxP
m
xW
1
11
1 ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
[ ]
( )
( )
−
−
=
−
−
−
+−
−
−
+
=
−+
−−
−
−
+
=
∑
∑
=
−
=+
−
=
−−
evennifn
oddnifnn
xP
m
nm
mn
x
x
xP
xP
rnr
rn
x
x
xPxQ
n
m
mnn
mr
n
r
rnnn
2/2
2/1
2
1
2
1
122
1
1
ln
2
1
12
142
1
1
ln
2
1
1
12
2
1
0
12
79. Legendre Ordinary Differential EquationSOLO
79
Legendre Functions of the Second Kind Qn (x)
( ) ( ) ( )xPxuxy nnn =
With Pn (x) being a solution of the Legendre Differential Equation
we look for the second solution having the form
( ) ( ) 0121 2
2
2
=++−− wnn
xd
wd
x
xd
wd
x
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
2
2
2
2
2
2
2
xd
xPd
xu
xd
xPd
xd
xud
xP
xd
xud
xd
xyd
xd
xPd
xuxP
xd
xud
xd
xyd
n
n
nn
n
nn
n
nn
nn
++=
+=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 01221121 2
2
22
2
2
2
=++−−−+−+− xPxunn
xd
xPd
xuxxP
xd
xud
x
xd
xPd
xxu
xd
xPd
xd
xud
xxP
xd
xud
x nn
n
nn
nn
n
nn
n
n
Substituting in the Legendre ODE we obtain
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 01212121
0
2
2
22
2
2
2
=
++−−+−−+− xuxPnn
xd
xPd
x
xd
xPd
xxP
xd
xud
x
xd
xPd
xd
xud
xxP
xd
xud
x nn
nn
n
nnn
n
n
or
80. SOLO
80
Legendre Functions of the Second Kind Qn (x) (continue – 1)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 02121 2
2
2
2
=−−+− xP
xd
xud
x
xd
xPd
xd
xud
xxP
xd
xud
x n
nnn
n
n
Equivalent to
( )
( )
( )
( )
0
1
2/
2
/
/
2
22
=
−
−+
x
x
xP
xdxPd
xdxud
xdxud
n
n
n
n
( ) ( ) ( ) 01lnln2ln 2
=−++ x
xd
d
xP
xd
d
xd
xud
xd
d
n
n
or
Integrating we obtain
( ) ( )[ ] ( ) .1lnlnln 22
constxxP
xd
xud
n
n
=−++
Therefore ( ) ( )[ ] ( ) AconstxxP
xd
xud
n
n
==−⋅⋅ .1 22
( )
( )[ ] ( )∫ −⋅
= 22
1 xxP
xd
Axu
n
n
This means that the second solution has the form
( ) ( ) ( ) ( )
( )[ ] ( )∫ −⋅
== 22
1 xxP
xd
xPxuxPxQ
n
nnnn
Legendre Ordinary Differential Equation
82. SOLO
82
Legendre Functions of the Second Kind Qn (x) (continue – 2)
( )
( ) ( )
( ) ( )
( ) ( )
3
2
2
5
1
1
ln
2
1
3
2
2
5
1
1
ln
4
35
2
3
1
1
ln
2
1
2
3
1
1
ln
4
43
1
1
1
ln
2
1
1
1
1
ln
2
1
1
ln
2
1
2
3
23
3
2
2
2
11
0
+−
−
+
=+−
−
+−
=
−
−
+
=−
−
+−
=
−
−
+
=−
−
+
=
−
+
=
x
x
x
xP
x
x
xxx
xQ
x
x
x
xP
x
x
xx
xQ
x
x
xP
x
xx
xQ
x
x
xQ
Legendre Ordinary Differential Equation
83. SOLO
83
Legendre Functions of the Second Kind Qn (x) (continue – 3)
To obtain a general formula for Qn (x), start from the Polynomial Pn (x) that has
n zeros αi, i=1,2,…,n
( ) ( ) ( ) ( )nnn xxxkxP ααα −−−= 21
( )[ ] ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )∑=
−
+
−
+
+
+
−
=
+⋅−⋅−−−
=
−⋅
n
i i
i
i
i
nnn
x
d
x
c
x
b
x
a
xxxxxkxxP
1
2
00
22
2
2
1
222
11
11
1
1
1
αα
ααα
( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( ) ( )∑=
−
+
−
−⋅+−++=
n
i i
i
i
i
nnn
x
d
x
c
xxPxPxbxPxa
1
2
222
0
2
0 1111
αα
If we put x=1 and x = -1, and remembering that Pn (1) = 1 and Pn(-1)=(-1)n
, we obtain
2/100 == ba
Let prove that
( )
( )[ ] ( )
0
1
1
22
2
=
−⋅
−=
= ixn
ii
xxP
x
xd
d
c
α
α
Legendre Ordinary Differential Equation
84. SOLO
84
Legendre Functions of the Second Kind Qn (x) (continue – 4)
Let prove that ( )
( )[ ] ( )
0
1
1
22
2
=
−⋅
−=
= ixn
ii
xxP
x
xd
d
c
α
α
Start with
( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) i
x
iixi xatfinitexfprovided
xd
xfd
xxfxxfx
xd
d
i
i
αααα
α
α ==
−+−=−
=
=
02
22
The only terms that are not finite in
at x = αi are the terms ci/(x-αi) and di/(x-αi)2
, therefore
( )[ ] ( ) ( ) ( )∑=
−
+
−
+
+
+
−
=
−⋅
n
i i
i
i
i
n x
d
x
c
x
b
x
a
xxP 1
2
00
22
111
1
αα
( )
( )[ ] ( )
( )
( ) ( )
( )[ ] i
x
iii
xi
i
i
i
i
xn
i cdxc
xd
d
x
d
x
c
x
xd
d
xxP
x
xd
d
iii
=
+−=
−
+
−
−=
−⋅
−
=== ααα
α
αα
αα 2
2
22
2
1
1
Therefore if we write Pn (x) = (x-αi) Li (x), we have
( )[ ] ( ) ( )[ ] ( )
( )
( )[ ] ( )
( ) ( ) ( )
( )[ ] ( )
( ) ( ) ( )
( )[ ] ( )22
2
223
2
2322222
1
/1
2
1
/122
1
/2
1
2
1
1
iii
iiiiii
xi
ii
xi
i
ixi
i
L
xdxLdL
xxL
xdxLdxxLx
xxL
xdxLd
xxL
x
xxLxd
d
c
iii
αα
αααα
ααα
−⋅
=−−
=
−⋅
−⋅−
=
−⋅
−
−⋅
=
−⋅
=
===
Legendre Ordinary Differential Equation
85. SOLO
85
Legendre Functions of the Second Kind Qn (x) (continue – 5)
We proved that ( )
( )[ ] ( ) ixn
ii
xxP
x
xd
d
c
α
α
=
−⋅
−= 22
2
1
1
( ) ( ) ( )
( )[ ] ( )22
2
1
/1
2
iii
iiiiii
i
L
xdxLdL
c
αα
αααα
−⋅
=−−
=Therefore if we write Pn (x) = (x-αi) Li (x), we have
Since Pn (x) = (x-αi) Li (x) satisfies the Legendre ODE, we have
( ) ( ) ( ){ } ( ) ( ){ } ( ) ( ) ( ){ } 0121 2
2
2
=−++−−−− xLxnnxLx
xd
d
xxLx
xd
d
x iiiiii ααα
Performing the calculation and substituting x = αi, we have
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 01221 2
2
2
=
−++
+−−
+−−
= ix
iii
i
i
ii
i xLxnnxL
xd
xLd
xx
xd
xLd
xd
xLd
xx
α
ααα
( ) ( ) ( ) 0212
2
=−
=
− iii
ii
i L
xd
xLd
αα
α
α
Substituting in ci equation we get
( ) ( ) ( )
( )[ ] ( ) 0
1
/1
2 22
2
=
−⋅
=−−
=
iii
iiiiii
i
L
xdxLdL
c
αα
αααα
Therefore ( )[ ] ( ) ( )∑= −
+
+
+
−
=
−⋅
n
i i
i
n x
d
xxxxP 1
222
1
1
1
1
2
1
1
1
α
Legendre Ordinary Differential Equation
86. SOLO
86
Legendre Functions of the Second Kind Qn (x) (continue – 6)
We can prove that but the exact value is not important, as
we shall see
( )
( )[ ] ( ) ixn
i
i
xxP
x
d
α
α
=
−⋅
−
= 22
2
1
Therefore
( )[ ] ( ) ( )∑= −
+
+
+
−
=
−⋅
n
i i
i
n x
d
xxxxP 1
222
1
1
1
1
2
1
1
1
α
( )[ ] ( ) ( ) ∑∑∫∫∫ == −
−
−
+
=
−
+
+
+
−
=
−⋅
n
i i
i
n
i i
i
n
x
d
x
x
dx
x
d
dx
xxxxP
dx
11
222
1
1
ln
2
1
1
1
1
1
2
1
1 αα
( ) ( ) ( )
∑= −
−
−
+
=
n
i i
n
inn
x
xP
d
x
x
xPxQ
11
1
ln
2
1
α
Since Pn (x)/(x-αi) is a polynomial of order (n-1) the sum above is also a
polynomial of order (n-1), and we define it as
( ) ( )
∑=
−
−
=
n
i i
n
in
x
xP
dxW
1
1 :
α
so
( ) ( ) ( ) 1
1
1
ln
2
1
1 <−
−
+
= − xxW
x
x
xPxQ nnn
Legendre Ordinary Differential Equation
87. SOLO
87
Legendre Functions of the Second Kind Qn (x) (continue – 7)
To find Wn-1 (x) let use the fact that Qn (x) is a solution of Legendre’s ODE
or
( ) ( ) ( ) ( ) ( ) 0121 2
2
2
=++−− xQnn
xd
xQd
x
xd
xQd
x n
nn
( ) ( ) ( ) ( )
xd
xWd
xP
xx
x
xd
xPd
xd
xQd n
n
nn 1
2
1
1
1
1
ln
2
1 −
−
−
+
−
+
=
( ) ( ) ( )
( )
( ) ( )
2
1
2
2222
2
2
2
1
2
1
2
1
1
ln
2
1
xd
xWd
xP
x
x
xd
xPd
xx
x
xd
xPd
xd
xQd n
n
nnn −
−
−
+
−
+
−
+
=
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0121
1
2
2
1
2
1
1
ln
2
1
121
1
1
2
1
2
2
22
0
2
2
2
=+−+−−
−
−+
−
+
−
+
++−−
−
−−
xWnn
xd
xWd
x
xd
xWd
xxP
x
x
xd
xPd
xP
x
x
x
x
xPnn
xd
xPd
x
xd
xPd
x
n
nn
n
n
n
n
nn
( ) ( ) ( ) ( ) ( ) ( )
xd
xPd
xWnn
xd
xWd
x
xd
xWd
x n
n
nn
2121 1
1
2
1
2
2
=++−− −
−−
Legendre Ordinary Differential Equation
( ) ( ) ( ) 1
1
1
ln
2
1
1 <−
−
+
= − xxW
x
x
xPxQ nnn
88. SOLO
88
Legendre Functions of the Second Kind Qn (x) (continue – 8)
Use the Recursive Formula
( ) ( ) ( ) ( ) ( )
xd
xPd
xWnn
xd
xWd
x
xd
d n
n
n
211 1
12
=++
− −
−
( ) ( ) ( ) ( )
( )
∑
−
=
−−−−=
1
2
1
0
121421
l
i
in
n
xPin
xd
xPd
Since Wn-1(x) is a polynomial of order n – 1, let write
( ) ( ) ( ) ( )
( )
∑
−
=
−−−−− =++=
1
2
1
0
1231101
n
i
ininnn xPaxPaxPaxW
Substitute those two equation in the Wn-1(x) O.D.E. to obtain
( ) ( ){ }
( )
( ) ( )
( )
( ) ( )
( )
∑∑∑
−
=
−−
−
=
−−
−
=
−− −−=++−
1
2
1
0
12
1
2
1
0
12
1
2
1
0
12
2
142211
l
i
in
n
i
ini
n
i
ini xPinxPannxPx
xd
d
a
But by Legendre O.D.E.: ( ) ( ){ } ( ) ( ) ( ) 02121 1212
2
=−−−+− −−−− xPininxPx
xd
d
inin
( ) ( ) ( ){ } ( )
( )
( ) ( )
( )
∑∑
−
=
−−
−
=
−− −−=++−−−−
1
2
1
0
12
1
2
1
0
12 14221212
l
i
in
n
i
ini xPinxPnninina
Let solve
Legendre Ordinary Differential Equation
89. SOLO
89
Legendre Functions of the Second Kind Qn (x) (continue – 9)
The coefficients of the same polynomial in both sides must be equal
( ) ( ) ( ){ } ( )
( )
( ) ( )
( )
∑∑
−
=
−−
−
=
−− −−=++−−−−
1
2
1
0
12
1
2
1
0
12 14221212
l
i
in
n
i
ini xPinxPnninina
( ) ( ) ( ){ } ( )14221212 −−=++−−−− innnininai
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )12224244
1221212
222
2
+−=−+−=++−+−+−=
++−+−−=++−−−−
iininininniniinn
nnininnninin
But
which gives
( ) ( )12
142
+−
−−
=
iin
in
ai
( )
( ) ( )
( )
( )
( ) ( ) ( ) ( )
[ ][ ]
∑ ∑∑ = =
−−
=+
−
=
−−−
+−
+−
=
−
−
+−
=
+−
−−
=
n
m
n
m
mnmn
mi
n
i
inn xP
mn
mn
m
xP
m
n
mn
m
xP
iin
in
xW
1 1
12
1
2
1
0
121
12
12221
2
1
1221
12
142
and
( ) ( ) ( )∑=
−−− =
n
m
mnmn xPxP
m
xW
1
11
1
?????
Legendre Ordinary Differential Equation
90. SOLO
90
Legendre Functions of the Second Kind Qn (x) (continue -10)
( )
3,2,1,0
10
=
<≤
n
xxQn
Legendre Ordinary Differential Equation
( )
( ) ( )
( ) ( )
( ) ( )
3
2
2
5
1
1
ln
2
1
3
2
2
5
1
1
ln
4
35
2
3
1
1
ln
2
1
2
3
1
1
ln
4
43
1
1
1
ln
2
1
1
1
1
ln
2
1
1
ln
2
1
2
3
23
3
2
2
2
11
0
+−
−
+
=+−
−
+−
=
−
−
+
=−
−
+−
=
−
−
+
=−
−
+
=
−
+
=
x
x
x
xP
x
x
xxx
xQ
x
x
x
xP
x
x
xx
xQ
x
x
xP
x
xx
xQ
x
x
xQ
91. SOLO
91
Legendre Functions of the Second Kind Qn (x) (continue -11)
Similar to Rodrigues Formula for Legendre Functions of the Second Kind Qn (x)
Start with
( )
( ) ( )12
2212
1
1
1
1 +−−
++
−=
−
n
nn
x
xx
( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( )( ) ( )
( )
( ) ( ) ( )( ) ( )1
32211
11,
,121,11,1:
++−
+−+−+−
−++=
−++=−+=−=
ini
nnn
uinnuf
unnufunufuuf
( )
( ) ( ) ( )
1
!!
!
!!
!1
1
1 12
0
2
0
2212
>
+
=
+
=
−
++−
∞
=
−
∞
=
++ ∑∑ xx
in
in
x
in
in
xx
in
i
i
i
nn
( )
( )
( ) ( )( ) ( ) 1
!
21
!
0
00
<
+++
== ∑∑
∞
=
∞
=
uu
i
innn
u
i
f
uf
i
i
i
i
i
Taylor expansion around u = 0
Use u = x-2
Legendre Ordinary Differential Equation
92. SOLO
92
Legendre Functions of the Second Kind Qn (x) (continue -12)
Similar to Rodrigues Formula for Legendre Functions of the Second Kind Qn (x)
Integrate relative to x
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( )
( )
∑∑
∫∑∫∑∫
∞
=
++−
∞
=
∞
++−
∞
++−
∞
=
∞
++−
∞
=
∞
+
−+
+
=
++
+
=
+
=
+
=
−
0
122
0
122
12
0
2
12
0
212
122!!
!
122!!
!
!
!
!
!
1
i
in
i
xin
x
in
ix
in
ix
n
x
inni
in
inni
in
d
n
in
d
n
ind
η
ηηηη
η
η
Integrate this n more times
( )
( )
( )
( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
( )
∑
∑
∑ ∫ ∫∫∫ ∫
∞
=
++−
∞
=
++−
∞
=
∞ ∞
++−
∞ ∞ ∞
+
+
++
++
=
+++++++
+
=
++
+
=
−
0
12
0
12
0
122
12
1
!122!!
!2!
122222122!!
!
122!!
!
1
i
in
i
in
i x
nin
x
n
n
x
inni
inin
x
ininininni
in
d
inni
ind
ηη η
ηη
η
η
Legendre Ordinary Differential Equation
93. SOLO
93
Legendre Functions of the Second Kind Qn (x) (continue -13)
Similar to Rodrigues Formula for Legendre Functions of the Second Kind Qn (x)
(continue)
( )
( ) ( )
( )
( )
1
!122!
!2!
!
1
1 0
12
12
1
>
++
++
=
−
∑∫∫ ∫
∞
=
++−
∞ ∞ ∞
+
+
xx
ini
inin
n
d
i
in
x
n
n
η η η
η
We found, using Frobenius Series
By comparing those two relations we obtain
( )
( )
1
1
!2 12
1
>
−
= ∫∫ ∫
∞∞ ∞
+
+
x
d
nxQ
x
n
n
n
n
η η η
η
Return to
Frobenius Series
( ) ( ) ( )
( )
( )
1
!122!
!2!
2
0
12
>
++
++
= ∑
∞
=
++−
xx
ini
inin
xQ
i
inn
n
Legendre Ordinary Differential Equation
94. SOLO
94
Legendre Functions of the Second Kind Qn (x) (continue -14)
Another Expression for Legendre Functions of the Second Kind Qn (x)
Start with the following Differential Equation
( ) ( ) 02121 2
2
2
=+−+− un
xd
ud
xn
xd
ud
x
One of the Solutions is . Check:( ) ( )n
xxu 12
1 −=
( ) ( ) ( ) ( ) 22212
2
1
2
121
11412&12
−−−
−−+−=−=
nnn
xxnnxn
xd
ud
xxn
xd
ud
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) 01211411412
2121
21221222
1
1
2
1
2
2
=−−−−+−−−−−=
+−+−
−− nnnn
xnxnxnxxnnxn
un
xd
ud
xn
xd
ud
x
The Second Solution can be find using u1 (x) by finding a function v (x) that satisfies:
( ) ( ) ( )n
xxvxu 12
2 −=
Legendre Ordinary Differential Equation
95. SOLO
95
Legendre Functions of the Second Kind Qn (x) (continue -15)
Another Expression for Legendre Functions of the Second Kind Qn (x)
(continue – 1)
( ) ( ) ( ) ( )
( ) ( ) 01221
21212121
112
2
1
12
0
1
1
2
1
2
2
2
2
2
2
2
2
=−+
+−+
+−+−=+−+−
xd
vd
uxnu
xd
vd
u
xd
vd
xd
ud
x
un
xd
ud
xn
xd
ud
xvun
xd
ud
xn
xd
ud
x
The Second Solution can be find using u1 (x) by finding a function v (x) that satisfies:
( ) ( ) ( )n
xxvxu 12
2 −=
2
1
2
1
12
2
2
2
2
1
1
2
2&
xd
ud
v
xd
ud
xd
vd
u
xd
vd
xd
ud
xd
ud
vu
xd
vd
xd
ud
++=+=
( )
( )
( )
( ) ( ) ( )
( )1
121
2
2
1
12
2
1
12
/
/
2
1
2
1
2
1
1
1
2
1
22
−
+
−=
−
−
−
−
=
−
−
−
−
=
x
xn
u
x
uxn
x
uxn
u
xd
ud
x
uxn
xdvd
xdvd
Legendre Ordinary Differential Equation
96. SOLO
96
Legendre Functions of the Second Kind Qn (x) (continue -16)
Another Expression for Legendre Functions of the Second Kind Qn (x)
(continue – 2)
Therefore
( )
( )1
12
/
/
2
22
−
+
−=
x
xn
xdvd
xdvd
( ) ( )
( )
( )
( )∫
∞
++
−
=⇒
−
=⇒−+−=
x
nn
d
xv
xxd
vd
xn
xd
vd
1212
2
11
1
1ln1
η
η
( ) ( ) ( ) ( )
( )∫
∞
+
−
−==
x
n
n d
xxuxvxu 12
2
12
1
1
η
η
Differentiate the Differential Equation ( ) ( ) 02121 2
2
2
=+−+− un
xd
ud
xn
xd
ud
x
( ) ( ) ( )
( ) ( ) ( ) 0122221
2121221
2
2
3
3
2
2
2
2
2
2
=−+−+−=
+−+−+−−
xd
ud
n
xd
ud
xn
xd
ud
x
xd
ud
n
xd
ud
xd
d
xn
xd
ud
n
xd
ud
x
xd
ud
xd
d
x
Legendre Ordinary Differential Equation
97. SOLO
97
Legendre Functions of the Second Kind Qn (x) (continue -17)
Another Expression for Legendre Functions of the Second Kind Qn (x)
(continue – 3)
Differentiate relative to x the Ordinary Differential Equation n times
( ) ( ) ( ) 0122221: 2
2
3
3
2
=−+−+−
xd
ud
n
xd
ud
xn
xd
ud
xODE
xd
d
( ) ( ) ( ) 0223321: 2
2
4
4
2
2
2
=−+−+−
xd
ud
n
xd
ud
xn
xd
ud
xODE
xd
d
( ) ( ) 02121: 2
2
2
=+−+− un
xd
ud
xn
xd
ud
xODE
( ) ( ) 0121: 1
1
2
2
2
=++−− +
+
+
+
n
n
n
n
n
n
n
n
xd
ud
nn
xd
ud
x
xd
ud
xODE
xd
d
Derive ( ) ( )[ ] ( )( ) 021121: 1
1
2
2
2
=−+++−+− +
+
+
+
i
i
i
i
i
i
i
i
xd
ud
ini
xd
ud
xin
xd
ud
xODE
xd
d
( ) ( )[ ]{ } ( )[ ] ( )( ){ }
( ) ( )[ ] ( )( ) 0122221
21121221:
1
1
1
1
3
3
2
1
1
1
1
3
3
2
=−−+++−+−
−+++−++−+−+−
+
+
+
+
+
+
+
+
+
+
+
+
i
i
i
i
i
i
i
i
i
i
i
i
i
i
xd
ud
ini
xd
ud
xin
xd
ud
x
xd
ud
iniin
xd
ud
xin
xd
ud
xODE
xd
d
xd
d
Proof
by
Induction q.e.d.
i = n
i → i+1
Legendre Ordinary Differential Equation
98. SOLO
98
Legendre Functions of the Second Kind Qn (x) (continue -18)
Another Expression for Legendre Functions of the Second Kind Qn (x) (continue – 4)
This is the Legendre Ordinary Differential Equation for dn
u/dxn
, having the solution
Pn (x) and Qn (x). Thus, the solution Pn (x) and Qn (x) can be written in the following
form:
( ) ( ) 0121: 1
1
2
2
2
=++−− +
+
+
+
n
n
n
n
n
n
n
n
xd
ud
nn
xd
ud
x
xd
ud
xODE
xd
d
We obtain
( ) ( )[ ]n
n
n
nn
n
nn x
xd
d
nxd
ud
n
xP 1
!2
1
!2
1 21
−==
( ) ( )
( )
( )
( )
( )
( )
1
1
1
!2
!21
!2
!21
12
22
>
−
−
−
=
−
= ∫
∞
+
x
d
x
xd
d
n
n
xd
ud
n
n
xQ
x
n
n
n
nnn
n
nnn
n
η
η
Rodrigues Formula
An integral for Qn (x) valid in |x| < 1 can be obtained from the previous result
( ) ( )
( )
( )
( )
1
1
1
!2
!21
0
12
2
<
−
−
−
= ∫ +
x
d
x
xd
d
n
n
xQ
x
n
n
n
nnn
n
η
η
Return to Table of Content
Legendre Ordinary Differential Equation
99. 99
SOLO
Laplace Differential Equation in Spherical Coordinates
0
sin
1
sin
sin
11
2
2
222
2
2
2
=
∂
Φ∂
+
∂
Φ∂
∂
∂
+
∂
Φ∂
∂
∂
=Φ∇
φθθ
θ
θθ rrr
r
rr
Let solve this equation by the method of Separation of Variables,
by assuming a solution of the form :
( ) ( )ϕθ,SrR=Φ
Spherical Coordinates:
θ
ϕθ
ϕθ
cos
sinsin
cossin
rz
ry
rx
=
=
=
In Spherical Coordinates the Laplace equation becomes:
Substituting in the Laplace Equation and dividing by Φ gives:
0sinsin
sin
11
2
2
22
2
2
=
∂
∂
+
∂
∂
∂
∂
+
φθ
θ
θ
θ
θ
SS
Srrd
Rd
r
rd
d
Rr
The first term is a function of r only, and the second of angular
coordinates. For the sum to be zero each must be a constant, therefore:
λ
φθ
θ
θ
θ
θ
λ
−=
∂
∂
+
∂
∂
∂
∂
=
2
2
2
2
sinsin
sin
1
1
SS
S
rd
Rd
r
rd
d
R
Associate Legendre Differential Equation
100. 100
SOLO
λ
φθ
θ
θ
θ
θ
−=
∂
∂
+
∂
∂
∂
∂
2
2
2
sinsin
sin
1 SS
S
( ) ( ) ( )φθφθ ,,, SrRr =Φ
We obtain:
Multiply this by S sin2
θ and put to get:( ) ( ) ( )φθφθ ΦΘ=,S
0
1
sinsinsin
1
2
2
2
=
Φ
Φ
+
+
Θ
∂
∂
Θ φ
θλ
θ
θ
θ
θ
d
d
d
d
Again, the first term, in the square bracket, and the last term must
be equal and opposite constants, which we write m2
, -m2
. Thus:
Φ−=
Φ
=Θ
−+
Θ
2
2
2
2
2
0
sin
sin
sin
1
m
d
d
m
d
d
d
d
φ
θ
λ
θ
θ
θθ
The Φ ( ) must be periodical in (a period of 2 π) and becauseϕ ϕ
this we choose the constant m2
, with m an integer. Thus:
( ) φφφ mbma sincos +=Φ
Laplace Differential Equation in Spherical Coordinates
( ) φφ
φ mjmj
ee +−
=Φ ,or
With m integer, we have the Orthogonality Condition
21
21
,
2
0
2 mm
mjmj
dee δπφ
π
φφ
=⋅∫
+−
Return to Table of Content
Associate Legendre Differential Equation
101. 101
SOLO
( ) ( ) ( )φθ ΦΘ=Φ rR
We obtain:
or:
Θ−=Θ−
Θ
+
Θ
λ
θθ
θ
θ 2
2
2
2
sin
cot
m
d
d
d
d
0
sin
sin
sin
1
2
2
=Θ
−+
Θ
∂
∂
θ
λ
θ
θ
θθ
m
d
d
Analysis of Associate Lagrange Differential Equation
Laplace Differential Equation in Spherical Coordinates
Change of variables: t = cos θ
θθ dtd sin−=
td
d
d
d Θ
−=
Θ
θ
θ
sin
( ) td
d
t
td
d
t
td
d
d
d
td
d
td
d
td
d
td
d
d
d
d
d
d
d Θ
−
Θ
−=
Θ
+
Θ
=
Θ
−−=
Θ
=
Θ
−
2
2
2
cossin/1
2
2
2
2
2
1
sin
sinsinsinsin
θθ
θ
θθ
θθθθ
θθθ
Θ
−
−+
Θ
−
Θ
−
Θ
=Θ
−+
Θ
+
Θ
=
=
2
2
2
2cos
2
2
2
2
1sin
cot0
t
m
td
d
t
td
d
t
td
dm
d
d
d
d t
λ
θ
λ
θ
θ
θ
θ
We obtain:
1cos
0
1
2 2
2
2
2
≤⇒=
=Θ
−
−+
Θ
−
Θ
tt
t
m
td
d
t
td
d
θ
λ
Associate Legendre Differential Equation
102. 102
SOLO
( ) ( ) ( )ϕθ gfrR=Φ
We obtain:
or:
ff
m
d
fd
d
fd
λ
θθ
θ
θ
−=−+ 2
2
2
2
sin
cot
Let try to factorize the left-hand side, second order differential
equation into two first-order operators:
0
sin
sin
sin
1
2
2
=
−+
∂
∂
f
m
d
fd
θ
λ
θ
θ
θθ
The two equations are identical if the coefficient are equal. This
is obtained by choosing α and β as follows:
( ) integer1
1
2
==−
=+
mmαβ
βα
( )
( ) ( ) ff
d
fd
d
fd
ff
d
fd
d
fd
f
d
fd
d
d
f
d
d
d
d
βα
θ
αβ
θ
θβα
θ
θβα
θ
β
θ
θβα
θ
θβ
θ
θα
θ
θβ
θ
θα
θ
θ
−−−++=
+−++=
+
+=
+
+
−
22
2
1
sin
1
2
22
2
sin
1
1cot
cot
sin
cot
cotcotcotcot
2
Laplace Differential Equation in Spherical Coordinates
Analysis of Associate Lagrange Differential Equation (continue – 1)
Associate Legendre Differential Equation
103. 103
SOLO
( ) ( ) ( )ϕθ gfrR=Φ
We obtain:
and:
ff
m
d
fd
d
fd
λ
θθ
θ
θ
−=−+ 2
2
2
2
sin
cot
We have two solutions for α and β as follows:
1.β1 = m, α1 = 1-m
2.β2 = -m, α2 = 1+m
Since m is an integer α, β are also integers.
( ) integer1
1
2
==−
=+
mmαβ
βα
( ) ( ) ( ) fff
d
fd
d
fd
λ
βαλβα
θ
αβ
θ
θβα
θ
+−=−−−++ '
sin
1
1cot 22
2
βα −=1
integer22
== mmβ
Let define the two operators:
( )
( )
( ) θ
θ
θ
θ
cot1
cot
++=
−=
−
+
m
d
d
M
m
d
d
M
m
m
Laplace Differential Equation in Spherical Coordinates
Analysis of Associate Lagrange Differential Equation (continue – 2)
Associate Legendre Differential Equation
104. 104
SOLO
( ) ( ) ( )ϕθ gfrR=Φ
We obtain:
We have two solutions for α and β as follows:
1.β1 = m, α1 = 1-m
2.β2 = -m, α2 = 1+m
Since m is an integer α, β are also integers.
( )
( )
( ) θ
θ
θ
θ
cot1
cot
++=
−=
−
+
m
d
d
M
m
d
d
M
m
m
ff
d
d
d
d
λθβ
θ
θα
θ
−=
+
+ cotcot
( )
( )
( )
( ) ( )11
11
11
11
1 mmmm fmmfMM
−−−=
−
−
−
+
−
αβ
βα
λ
We have:
( )
( )
( )
( ) ( )22
22
22
1 mmmm fmmfMM
+−−=
−
+−
αβ
βα
λ
fm
(1)
– the solution for α1, β1
fm
(2)
– the solution for α2, β2
Laplace Differential Equation in Spherical Coordinates
Analysis of Associate Lagrange Differential Equation (continue – 3)
Associate Legendre Differential Equation
105. 105
SOLO
( ) ( ) ( )ϕθ gfrR=Φ
We obtain:
( ) ( ) ( )
( )[ ] ( )11
11 1 mmmm fmmfMM −−−=−
−
+
− λ
( ) ( ) ( )
( )[ ] ( )22
1 mmmm fmmfMM +−−=+−
λ
Let operate on first of those equations with
( )−
−1mM
( ) ( ) ( ) ( )
[ ] ( )[ ] ( ) ( )
[ ]1
1
1
111 1 mmmmmm fMmmfMMM −
−
−
−
+
−
−
− −−−= λ
In the Second Equation replace m by m-1
( ) ( ) ( )
( )[ ] ( )2
1
2
111 1 −−
+
−
−
− −−−= mmmm fmmfMM λ
Let operate on second of those equations with
( )+
mM
1
2
( ) ( ) ( ) ( )
[ ] ( )[ ] ( ) ( )
[ ]22
1 mmmmmm fMmmfMMM ++−+
+−−= λ
In the First Equation replace m by m+1
( ) ( ) ( )
( )[ ] ( )1
1
1
1 1 ++
−+
+−−= mmmm fmmfMM λ
( ) ( ) ( )21
1 mmmm fpfM =+
−
( ) ( ) ( )1
1
2
+
+
= mmmm fqfM
Laplace Differential Equation in Spherical Coordinates
pm is a constant
qm is a constant
mm →−1
mm →−1
Analysis of Associate Lagrange Differential Equation (continue – 4)
Associate Legendre Differential Equation
106. 106
SOLO
We
obtained:( ) ( ) ( )21
1 mmmm fpfM =+
−
( ) ( ) ( )1
1
2
+
+
= mmmm fqfM
Laplace Differential Equation in Spherical Coordinates
( )
( )
( )
θ
θ
θ
θ
cot
cot1
m
d
d
M
m
d
d
M
m
m
−=
++=
+
−
where:
Theorem:
( ) ( )
( ) ( ) ( )
( )∫∫
+−
−=
ππ
θθθθθθθθ
00
sinsin dgMfdfMg mm
where f and g are arbitrary bounded function of θ.
Proof:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( )
( )∫∫
∫
∫∫
+
−
−=
−−=
+−−=
++=
ππ
π
π
ππ
θθθθθθθ
θ
θ
θθ
θ
θθ
θθθθθθ
θ
θθθθ
θθθ
θ
θθθθθθ
00
0
0
0
00
sinsin
sin
cos
sin
cos1sinsin
cot1sinsin
dgMfdgfmg
d
d
f
dgfmg
d
d
ffg
dfm
d
d
gdfMg
m
m
Those are Recursive Relations in m.
Analysis of Associate Lagrange Differential Equation (continue – 5)
Associate Legendre Differential Equation
108. 108
SOLO
Laplace Differential Equation in Spherical Coordinates
( ) ( )
( )( ) ( ) ( )
( )( )∫∫
+−
−=
ππ
θθθθθθθθ
00
sinsin dgMfdfMg mm
Choose now f := Mm
(+)
fm and g := fm, to obtain as before
We obtained
( ) ( ) λλ ≤+→+−= 11 mmmmqm
( ) ( ) λλ ≤+→+−= 11 mmmmpm
We see that m, an integer, is bounded by λ, therefore we must choose λ as
( ) 0integer1 >=+= lllλ
In this case we have mMAX = l, mmin = -(l+1), or
( ) integer0,integer1 =>=≤≤+− mllml
Therefore
( ) ( ) ( ) lmlmmllqp mm ≤≤+−→+−+== 111
Analysis of Associate Lagrange Differential Equation (continue – 7)
Associate Legendre Differential Equation
109. 109
SOLO
Laplace Differential Equation in Spherical Coordinates
We obtained
( ) ( ) ( ) lmlmmllqp mm ≤≤+−→+−+== 111
( )
( )
( )
θ
θ
θ
θ
cot
cot1
m
d
d
M
m
d
d
M
m
m
−=
++=
+
−
( ) ( ) ( )21
1 mmmm fpfM =+
−
( ) ( ) ( )1
1
2
+
+
= mmmm fqfM
( ) ( )
( ) ( ) ( )
( )
( ) ( )
( ) lml
mmllfm
d
d
fmmllfm
d
d
m
mm
≤≤+−
+−+=
−
+−+=
++ +
1
11cot
11cot1
2
21
1
θ
θ
θ
θ
Substituting m = - (l+1) in the First Equation and m = l in the Second we obtain:
( )
( )
=
−
=
− −
0cot
0cot
2
1
l
l
fl
d
d
fl
d
d
θ
θ
θ
θ
( )
( )
( )
( )
( )
( )
θ
θ
θ
θ
sin
sin
sin
sin
2
2
1
1
d
l
f
fd
d
l
f
fd
l
l
l
l
=
=
+
+
−
−
( ) ( )
θl
lll Cff sin
21
== +−
If we can find a Solution for a particular m, we can use the Recursive Relations
above to find the others.
Analysis of Associate Lagrange Differential Equation (continue – 8)
Associate Legendre Differential Equation
110. 110
SOLO
Laplace Differential Equation in Spherical Coordinates
We obtained ( ) ( )
θl
lll Cff sin
21
== +−
Cl must be chosen to normalize fi and f-l:
1sinsin
0
122
0
2
== ∫∫
+
±
ππ
θθθθ dCdf l
ll
( )
( )!12
!21
212
2
+
=
+
n
n
C
n
l
( )
( )212
!2
!12
n
n
C nl +
+
=
( ) ( ) ( )∫∫∫∫
+−−+
−=+−=−=
ππ
π
ππ
θθθθθθθθθθθθ
0
1212
0
212
0
0
2
0
2
0
12
sinsin2cossin2cossincossinsin dndndd nnnnnn
Therefore
( )
( ) ( ) ( ) ( ) ( )
( )
( )!12
!2
cos
2222
!2
11212
!2
sin
31212
2222
sin
12
2
sin
212
2
0
1
00
12
0
12
+
=
−⋅
⋅
−⋅+
=
−⋅+
−⋅
=
+
=
+
−+
∫∫∫ n
n
nn
n
nn
n
d
nn
nn
d
n
n
d
nnn
nn
π
πππ
θθθθθθθ
Analysis of Associate Lagrange Differential Equation (continue – 9)
Associate Legendre Differential Equation
111. 111
SOLO
Laplace Differential Equation in Spherical Coordinates
The Normalized Solution for m = l is defined as
( )
( )
θl
n
lm
l
lm
l
n
n
sin
!2
!12
212 +
−== +
=Θ=Θ
The Solutions for m < l can be found using the Recursive Relation:
( )
( ) ( )
( )1,,,0,,1,cot
11
11
1
1
1
+−−−=Θ
+
−−+
=Θ=Θ −
−
−
−
llllmm
d
d
mmll
M
p
m
l
m
lm
m
m
l θ
θ
( ) ( ) ( )21
1 mmmm fpfM =+
−
From which the Normalized Solutions for m < l are given by
Or we can find the Solutions for m >- l by using the Recursive Relation:
( ) ( ) ( )1
1
2
+
+
= mmmm fqfM
( )
( ) ( )
llllmm
d
d
mmll
M
q
m
l
m
lm
m
m
l ,1,,0,,,1cot
11
111
−−−−=Θ
−
+−+
=Θ=Θ ++
θ
θ
Analysis of Associate Lagrange Differential Equation (continue – 10)
Associate Legendre Differential Equation
112. 112
SOLO
Laplace Differential Equation in Spherical Coordinates
( )
( )
θl
n
lm
l
lm
l
n
n
sin
!2
!12
212 +
−== +
=Θ=Θ
( )
( ) ( )
( )1,,,0,,1,cot
11
11
1
1
1
+−−−=Θ
+
−−+
=Θ=Θ −
−
−
−
llllmm
d
d
mmll
M
p
m
l
m
lm
m
m
l θ
θ
Examples:
( )
( ) ( )
llllmm
d
d
mmll
M
q
m
l
m
lm
m
m
l ,1,,0,,,1cot
11
111
−−−−=Θ
−
+−+
=Θ=Θ ++
θ
θ
Analysis of Associate Lagrange Differential Equation (continue – 11)
2
cos
1
1
cos
0
1
2
cos
1
1
0
0
1
2
3
sin
2
3
2
3
cos
2
3
1
2
3
sin
2
3
1
2
1
0
x
x
xl
l
x
x
x
−−=−=Θ
==Θ
−==Θ=
=Θ=
=
−
=
=
θ
θ
θ
θ
θ
θ
( )
( ) ( )
( )2
cos
22
2
2
cos
1
2
2
cos
20
2
2
cos
1
2
2
cos
22
2
1
4
15
sin
4
15
1
2
15
cossin
2
15
13
22
5
1cos3
22
5
1
2
15
cossin
2
15
1
4
15
sin
4
15
2
x
xx
x
xx
xl
x
x
x
x
x
−−=−=Θ
−−=−=Θ
−=−=Θ
−==Θ
−==Θ=
=
−
=
−
=
=
=
θ
θ
θ
θ
θ
θ
θθ
θ
θθ
θ
Return to Table of Content
Associate Legendre Differential Equation
113. SOLO
113
Associated Legendre Functions
Let Differentiate this equation m times with respect to t, and use
Leibnitz Rule of
Product Differentiation: ( ) ( )[ ]
( )
( ) ( )
im
im
i
im
i
m
m
td
tgd
td
tsd
imi
m
tgts
td
d
−
−
=
∑ −
=⋅
0 !!
!
Start with: ( ) ( ) ( ) ( ) 1011 2
≤=++
− ttwnntw
td
d
t
td
d
nn
or: ( ) ( ) ( ) ( ) ( ) 10121 2
2
2
≤=++−− ttwnntw
td
d
ttw
td
d
t nnn
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( )twmmtwtmtwttw
td
d
t
td
d m
n
m
n
m
nnm
m
1211
122
2
2
2
−−−−=
−
++
( ) ( )
( ) ( )
( )twmtwttw
td
d
t
td
d m
n
m
nnm
m
+=
+1
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( )twnntwmtwttwmmtwtmtwt
m
n
m
n
m
n
m
n
m
n
m
n 122121
1122
++−−−−−−
+++
( ) ( )
( ) ( ) ( )
( ) ( ) ( )[ ] ( )
( ) 011121
122
=+−+++−−=
++
twmmnntwtmtwt
m
n
m
n
m
n
2nd
Way
114. SOLO
114
Associated Legendre Functions
( ) ( )
( ) ( ) ( )
( ) ( ) ( )[ ] ( )
( ) 011121
122
=+−+++−−
++
twmmnntwtmtwt
m
n
m
n
m
n
Define: ( ) ( )
( )twty
m
n=:
( ) ( )
( ) ( ) ( )
( ) ( ) ( )[ ] ( ) 011121 122
=+−+++−− tymmnntytmtyt
Now define: ( ) ( ) ( )tyttu
m
22
1: −=
Let compute:
( ) ( ) ( )1221
22
11 ytyttm
td
ud mm
−+−−=
−
( ) ( ) ( ) ( )11
22222
111 ytyttm
td
ud
t
mm
+
−+−−=−
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )21
221221221
2222222
1121111 ytyttmyttmyttmytm
td
ud
t
td
d mmmmm
+−
−+−+−−−−+−−=
−
( ) ( ) ( )
( ) ( )
( ) ( )[ ]{ } ( ) ( ) y
t
tm
mnnmmtymmnnytmytt
mm
−
+−+−+−++−+++−−−= 2
22
222
0
12222
1
11111211
We get: ( ) ( ) 0
1
11 2
2
2
=
−
−++
− u
t
m
nn
td
ud
t
td
d
2nd
Way
115. SOLO
115
Associated Legendre Functions
Define: ( ) ( ) ( )tw
td
d
ttu nm
mm
22
1: −=
We get: ( ) ( ) 0
1
11 2
2
2
=
−
−++
− u
t
m
nn
td
ud
t
td
d
Start with Legendre Differential Equation:
( ) ( ) ( ) ( ) 1011 2
≤=++
− ttwnntw
td
d
t
td
d
nn
Summarize
But this is the Differential Equation of f (θ) obtained by solving Laplace’s Equation
by Separation of Variables in Spherical Coordinates .
02
=Φ∇
( ) ( ) ( ) ( )φθφθ ΦΘ=Φ rRr ,,
The Solutions of this Differential Equation are called Associated Legendre
Functions, because they are derived from the Legendre Polynomials, and
Legendre Functions of the Second Kind Qn (x) and are denoted:
( ) ( ) ( )tP
td
d
ttP nm
mm
m
n
22
1: −=
2nd
Way
( ) ( ) ( )tQ
td
d
ttQ nm
mm
m
n
22
1: −=
116. SOLO
116
Associated Legendre Functions
Let use Rodrigues Formula for Pn (t):
We see that we can define the Associated Legendre Function even for negative
m (In the Differential equation m2
appears):
( ) ( ) ( )tP
td
d
ttP nm
mm
m
n
22
1: −=
( ) ( )[ ]n
n
n
nn t
td
d
n
tP 1
!2
1 2
−=
we obtain:
Associated Legendre Functions
( ) ( ) ( )[ ]n
mn
mnm
n
m
n t
td
d
t
n
tP 11
!2
1
: 222
−−= +
+
From this equation we obtain:
( ) ( )tPtP nn =
0
2nd
Way
Hinweis der Redaktion
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
Nagle, Saff, Snider, “Fundamentals of Differential Equations and Boundary Value Problems“, 4th Ed., Pearson/Addison Wesley, 2004, Ch. 11, “Eigenvalue Problems and Sturm-Liouville Equations”
http://en.wikipedia.org/wiki/Legendre_polynomials
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
K.T. Tang, “Mathematical Methods for Engineers and Scientists”, Vol 3, Springer-Verlag, 2007
http://en.wikipedia.org/wiki/Ferdinand_Georg_Frobenius
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Wallace, P.R., “Mathematical Analysis of Physical Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of Physical Problems”, Dover 1972, 1984
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
L.C. Andrews, “Special Functions for Engineers and Applied Mathematics”, MacMillan Publishing Companie, 1985
Afken, Weber, “Mathematical Methods for Physicists”, Harcourt Academic Press, 5th Ed., 2001
http://en.wikipedia.org/wiki/Franz_Ernst_Neumann
http://journals.cambridge.org/action/displayAbstract;jsessionid=4BE79E68247E19A6FFC459F00AAFE923.journals?fromPage=online&aid=2061924
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
http://en.wikipedia.org/wiki/Legendre_polynomials
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
“Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, M. Abramowitz, I. Stegun
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
W. W. Bell, “Special Functions for Scientists and Engineers”, D. van Nostrand Company ltd., 1968
Arfken, Weber, “Mathematical Methods for the Physicists”, 5 Ed., Harcourt – Academic Press, 2001
“Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, M. Abramowitz, I. Stegun
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
S.I. Hayek, “Advanced Mathematical Methods in Science and Engineering”, Marcel Dekker Inc., 2001
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984
Wallace, P.R., “Mathematical Analysis of PhysicaL Problems”, Dover 1972, 1984