Chapter 21 - Legendre Polynomials - DSP-Book
Chapter 21 - Legendre Polynomials - DSP-Book
Chapter 21 - Legendre Polynomials - DSP-Book
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Poularikas A. D. “<strong>Legendre</strong> <strong>Polynomials</strong>”<br />
The Handbook of Formulas and Tables for Signal Processing.<br />
Ed. Alexander D. Poularikas<br />
Boca Raton: CRC Press LLC,1999<br />
© 1999 by CRC Press LLC
<strong>21</strong>.1 <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.1.1 Definition<br />
<strong>21</strong>.1.2 Generating Function<br />
© 1999 by CRC Press LLC<br />
w(–t,–s) = w(t,s)<br />
<strong>21</strong>.1.3 Rodrigues Formula<br />
<strong>21</strong>.1.4 Recursive Formulas<br />
1.<br />
2.<br />
P () t =<br />
n<br />
<strong>21</strong><br />
<strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.1 <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.2 <strong>Legendre</strong> Functions of the Second Kind<br />
(Second Solution)<br />
<strong>21</strong>.3 Associated <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.4 Bounds for <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.5 Table of <strong>Legendre</strong> and Associate <strong>Legendre</strong><br />
Functions<br />
References<br />
[ n / 2] k n 2k<br />
∑<br />
k=<br />
0<br />
−<br />
( −1) ( 2n−2k)! t<br />
n<br />
2 k!( n−k)!( n−2k)! ⎧n/<br />
2 neven<br />
[ n / 2]<br />
= ⎨<br />
⎩(<br />
n−1)/ 2 n odd<br />
wts (, ) =<br />
1<br />
1− 2st<br />
+ s<br />
2<br />
⎧<br />
⎪<br />
⎪<br />
= ⎨<br />
⎪<br />
⎪<br />
⎩<br />
∞<br />
∑<br />
n=<br />
0<br />
∞<br />
∑<br />
n=<br />
0<br />
n<br />
P () t s s < 1<br />
n<br />
−n−1 P () t s s > 1<br />
n<br />
1 d 2 n<br />
Pn() t = ( t − 1) n = 012 , , L<br />
n n<br />
2 n!<br />
dt<br />
( n+ 1) P () t − ( 2n+ 1) tP() t + nP () t = 0 n = 12 , ,<br />
n+ 1 n n−1<br />
L<br />
P′ + − ′ = + ′ = ⋅<br />
n 1() t tPn() t ( n 1) Pn() t ( P () t derivative of P() t n = 012 ,, , L<br />
n<br />
generating function
3.<br />
4.<br />
5.<br />
6.<br />
Figure <strong>21</strong>.1 shows a few <strong>Legendre</strong> functions.<br />
FIGURE <strong>21</strong>.1<br />
<strong>21</strong>.1.5 <strong>Legendre</strong> Differential Equation<br />
If y = P ( x) ( n = 012L , , , ) is a solution to the second-order DE<br />
For<br />
tP′ () t − P′ () t = nP() t n=<br />
12L , ,<br />
n n−1 n<br />
Pn′ + 1() t − Pn′ −1()<br />
t = ( 2n+ 1) Pn() t n = 12 , ,L<br />
2<br />
( t − 1)<br />
Pn′ () t = ntPn() t −nPn−1() t<br />
P() t = 1 P() t = t<br />
n<br />
0 1<br />
t = cos ϕ:<br />
© 1999 by CRC Press LLC<br />
TABLE <strong>21</strong>.1 <strong>Legendre</strong> <strong>Polynomials</strong><br />
P 0<br />
= 1<br />
P1 t =<br />
3 P = t −<br />
2<br />
2<br />
2 1 2<br />
5 P = t − t<br />
3<br />
2<br />
3 3 2<br />
35 P = t − t +<br />
4<br />
8<br />
4 30 8<br />
2 3 8<br />
63 P = t − t + t<br />
5<br />
8<br />
5 70 8<br />
3 15 8<br />
231 6 315 4 105<br />
P = t − t + t −<br />
6<br />
16<br />
16<br />
16<br />
2 5<br />
16<br />
429 7 693 5 315 3 35<br />
P = t − t + t − t<br />
7<br />
16<br />
16<br />
16<br />
2<br />
( 1− t ) y′′ − 2ty′ + n( n+ 1) y = 0<br />
1 d ⎛ dy ⎞<br />
⎜sin<br />
ϕ ⎟ + nn ( + 1) y=<br />
0<br />
sinϕ<br />
dϕ⎝<br />
dϕ⎠<br />
16
Example<br />
From (<strong>21</strong>.1.4.4) and t = 1 implies 0 = nPn() 1 − nPn−1() 1 or Pn() 1 = Pn−1().<br />
1 For n = 1, P1( 1) = P0(<br />
1) = 1.<br />
For n = 2, P ( 1) = P(<br />
1) = 1 and so forth. Hence Pn () 1 = 1.<br />
<strong>21</strong>.1.6 Integral Representation<br />
1. Laplace integral:<br />
2. Mehler-Dirichlet formula:<br />
3. Schläfli integral:<br />
1<br />
Pn() t =<br />
2π<br />
j<br />
C<br />
2 n<br />
( z − 1)<br />
dz<br />
n n+<br />
1<br />
2 ( z−t) where C is any regular, simple, closed curve surrounding t.<br />
<strong>21</strong>.1.7 Complete Orthonormal System<br />
The <strong>Legendre</strong> polynomials are orthogonal in [–1,1]<br />
and therefore the set<br />
is orthonormal.<br />
2 1<br />
<strong>21</strong>.1.8 Asymptotic Representation:<br />
δ = fixed positive number<br />
<strong>21</strong>.1.9 Series Expansion<br />
If f(t) is integrable in [–1,1] then<br />
© 1999 by CRC Press LLC<br />
∫<br />
π<br />
1<br />
2<br />
n<br />
Pn( t) = [ t+ t −1cos<br />
ϕ] dϕ<br />
π 0<br />
2<br />
Pn<br />
(cos θ)<br />
=<br />
π<br />
1<br />
1/ 2<br />
{[ ( 2n+ 1)]<br />
P ( t)}<br />
∫<br />
2<br />
n m<br />
−1<br />
∫<br />
1<br />
1<br />
−1<br />
P () t P () t dt = 0<br />
n<br />
θ<br />
∫0<br />
2 2<br />
[ Pn( t)] dt =<br />
n = 012 , , L<br />
2n+ 1<br />
2n+ 1<br />
() P () t n = 012L ,,<br />
2<br />
ϕn t =<br />
n<br />
∫<br />
1 cos( n + 2 ) ψ<br />
dψ 0< θ< π,<br />
n = 012 , , , …<br />
2cosψ−cosθ<br />
Pn<br />
(cos )<br />
sin n , n<br />
nsin<br />
θ<br />
π<br />
≅<br />
⎛<br />
+<br />
⎞<br />
θ<br />
δ θ π δ<br />
π θ ⎝ ⎠ +<br />
2 ⎡ 1 ⎤<br />
⎢<br />
⎥ →∞, ≤ ≤ −<br />
⎣ 2 4⎦<br />
∞<br />
∑<br />
n=<br />
0<br />
f() t = a P () t − 1< t < 1<br />
n n
For even f(t), the series will contain term Pn(t) of even index; if f(t) is odd, the term of odd index only.<br />
If the real function f(t) is piecewise smooth in (–1,1) and if it is square integrable in (–1,1), then the<br />
series converges to f(t) at every continuity point of f(t). If there is a discontinuity at t then the series<br />
converges at [ f( t + 0) + f( t −0)]/<br />
2.<br />
<strong>21</strong>.1.10 Change of Range<br />
If a function f(t) is defined in [a,b], it is sometimes necessary in the applications to expand the function<br />
in a series in the applications to expand the function in a series of orthogonal polynomials in this interval.<br />
Clearly the substitution<br />
transform the interval [a,b] of the x-axis into the interval [–1,1] of the t-axis. It is, therefore, sufficient<br />
to consider<br />
The above equation can also be accomplished as follows:<br />
Example<br />
Suppose f(t) is given by<br />
Then<br />
© 1999 by CRC Press LLC<br />
a<br />
2n+ 1<br />
=<br />
2<br />
∫<br />
n n<br />
1<br />
−1<br />
f () t P () t dt n = 012 ,, L<br />
t<br />
b a x<br />
b a<br />
b a b a<br />
= − a b x t<br />
−<br />
+ ⎡ ⎤<br />
−<br />
< = +<br />
⎣⎢ ⎦⎥<br />
+<br />
2<br />
, ,<br />
⎡<br />
⎤<br />
2 ⎣⎢ 2 2 ⎦⎥<br />
f b a − b a<br />
t + +<br />
⎡<br />
⎤<br />
⎣⎢<br />
⎦⎥ =<br />
2 2<br />
a<br />
1<br />
∫<br />
∞<br />
∑<br />
n=<br />
0<br />
a P () t<br />
n n<br />
n<br />
f b a + − b a<br />
=<br />
t + P t dt<br />
+<br />
2 1 ⎡<br />
⎤<br />
()<br />
2 ⎣⎢ 2 2 ⎦⎥<br />
n n<br />
−1<br />
f() t = a X () t<br />
1<br />
X () t =<br />
n!( b−a) a<br />
n<br />
∞<br />
∑<br />
n=<br />
0<br />
2n+ 1<br />
=<br />
b−a n n<br />
n n<br />
a<br />
∫<br />
b<br />
d ( t −a) ( t −b)<br />
n<br />
dt<br />
n n n<br />
f() t X () t dt<br />
⎧0<br />
−1≤ t < a<br />
f()= t ⎨<br />
⎩1<br />
a < t ≤1<br />
a<br />
n<br />
2n+ 1<br />
=<br />
2<br />
1<br />
∫<br />
a<br />
n<br />
P ()<br />
t dt<br />
n
Using (<strong>21</strong>.1.4.4), and noting that P n(1) =1 we obtain<br />
which leads to the expansion<br />
Example<br />
Suppose f(t) is given by<br />
The function is an odd function and, therefore, f(t)P n(t) is an odd function of P n(t) with even index.<br />
Hence a n are zero for n = 0,2,4,…. For odd index n, the product f(t)P n(t) is even and hence<br />
Using (<strong>21</strong>.1.4.4) and setting n = 2k + 1, k = 0,1,2… we obtain<br />
where we have used the property P n(1) =1 for all n. But<br />
and, thus, we have<br />
The expansion is<br />
© 1999 by CRC Press LLC<br />
1<br />
1<br />
a =− [ P ( a) − P ( a)], a = ( −a)<br />
n 2 n+ 1 n−1<br />
0<br />
∞<br />
∑<br />
1<br />
f() t ≅ ( −a) − [ Pn+ ( a) − Pn− ( a)] Pn(), t − < t <<br />
2 1<br />
1<br />
1 1<br />
1 1<br />
2<br />
1<br />
n=<br />
1<br />
2 1<br />
⎧−1<br />
−1≤ t < 0<br />
f()= t ⎨<br />
⎩1<br />
0 < t ≤1<br />
∫ ∫<br />
an =<br />
⎛ 1<br />
n+ ⎞<br />
f t Pn t dt =<br />
⎛ 1<br />
() () 2 n+ ⎞<br />
Pn() t dt n = 135 , , ,L<br />
⎝ 2⎠<br />
⎝ 2⎠<br />
a<br />
−1<br />
1<br />
∫ ∫<br />
a = ( 4k+ 3)<br />
P ( t) dt = [ P′ ( t) − P′ ( t)] dt<br />
2k+ 1<br />
2k+ 1<br />
2k+ 2 2k<br />
0<br />
0<br />
1<br />
= [ P ( t) − P ( t)] = P ( 0) −P<br />
( 0)<br />
2k+ 2 2k 0 2k 2k+ 2<br />
P<br />
2n<br />
1<br />
n<br />
2 ( 1) ( 2n)!<br />
( 0)<br />
= 2n2 n 2 (!) n<br />
− ⎛ ⎞<br />
⎜ ⎟ =<br />
⎝ ⎠<br />
−<br />
k<br />
k+<br />
1<br />
k<br />
1 2k<br />
1 2k2 1 2k<br />
2k1 = 1<br />
k<br />
2k+ 2 2 2k 2<br />
2 k 2 k 1 2 k 2k2 − − +<br />
− =<br />
+<br />
−<br />
( ) ( )! ( ) ( ) ( ) ( )! ⎡ +<br />
+<br />
⎤<br />
(!) [( )!] (!) ⎣⎢ + ⎦⎥<br />
2k+ 1 2 2<br />
k<br />
1 2k 4k 3<br />
= 2k+ 1<br />
2 k k 1<br />
− +<br />
( ) ( )!( )<br />
!( + )!<br />
f() t =<br />
∞<br />
∑<br />
n=<br />
0<br />
1<br />
1<br />
0<br />
k<br />
( − 1) ( 2k)!( 4k + 3)<br />
P t t<br />
k k () − ≤ ≤<br />
2 + 1 2 + 1 1 1<br />
2 k!( k + 1)!
<strong>21</strong>.1.11 Expansion of <strong>Polynomials</strong><br />
k<br />
If q t c x is an arbitrary polynomial, then where =<br />
m()= k<br />
qm() t = c0P0() t + c1P1() t + + cmPm() t L cn ∫<br />
Example<br />
To find P 2n(0) we use the summation<br />
with k = 0. Hence<br />
Example<br />
m<br />
∑<br />
k=<br />
0<br />
1<br />
⎛ 1<br />
n+ ⎞<br />
qm() t Pn() t dt = 0, n = 012 ,, L.<br />
⎝ 2⎠<br />
−1<br />
1<br />
q t P t dt = 0 m < r<br />
∫ m() r()<br />
, .<br />
−1<br />
∫<br />
© 1999 by CRC Press LLC<br />
1<br />
n k<br />
P t<br />
k n k n k t<br />
n<br />
( 1)<br />
( 1) ( 2 2 1)!<br />
2n () = 2n1 2 ( 2 )!( 1)!(<br />
)!<br />
− − + −<br />
−<br />
+ − −<br />
P<br />
If q m(t) is a polynomial of degree m and m < r, then<br />
To evaluate P for m ≠ 0 we must consider the two cases: m = odd and m = even.<br />
m t dt<br />
0<br />
(a) m = even and m ≠ 0<br />
()<br />
The result is due to the orthogonality principle.<br />
(b) m = odd and m ≠ 0. From the relation (see Table <strong>21</strong>.2)<br />
with t = 0 we obtain<br />
∑<br />
n k<br />
k=<br />
0<br />
n<br />
n<br />
n<br />
( 1) ( 2n1)! ( 1) 2n[( 2n 1)!]<br />
( 1) ( 2n)!<br />
( 0)<br />
= n<br />
n<br />
n<br />
2 ( n 1)!<br />
n!<br />
2 n[( n 1)!]<br />
n!<br />
2 (!) n<br />
− −<br />
=<br />
−<br />
− −<br />
=<br />
−<br />
−<br />
−<br />
2n 2 1 2 2 2<br />
Using the results of the previous example, we obtain<br />
1<br />
∫ 0<br />
2k<br />
1<br />
1<br />
1<br />
1<br />
1<br />
1<br />
1<br />
1<br />
∫ m<br />
0 ∫ m<br />
1 ∫ m ⋅<br />
− −1 ∫ m<br />
−1<br />
P () t dt = P () t dt = P () t dt = P () t P0() t dt = 0<br />
2<br />
2<br />
2<br />
∫<br />
1<br />
1<br />
P () t dt = [ P () t − P ()] t<br />
2m+ 1<br />
m m− 1 m+<br />
1<br />
−1<br />
1<br />
∫ 0<br />
1<br />
Pm() t dt = [ Pm− 1( 0) − Pm+<br />
1(<br />
0)]<br />
2m+ 1<br />
⎡<br />
⎤<br />
⎢ m− 1<br />
m+<br />
1 ⎥<br />
2<br />
2<br />
1 1 1 1 1<br />
= ⎢ ( − ) ( m − )! ( − ) ( m + )!<br />
P 2 − ⎥<br />
m()<br />
t dt<br />
2<br />
2m+ 1⎢<br />
1⎡<br />
−1<br />
1 1<br />
2<br />
⎛ ⎞ ⎤ ⎡ +<br />
⎢<br />
2<br />
⎛ ⎞ ⎤ ⎥<br />
m− m<br />
m+<br />
m<br />
⎢<br />
!<br />
⎣⎝<br />
2 ⎠ ⎥ ⎢ !<br />
⎦ ⎣⎝<br />
2 ⎠ ⎥ ⎥<br />
⎣<br />
⎦ ⎦<br />
m−1<br />
m−1<br />
2<br />
2<br />
( −1) ( m− 1)!( 2m+ 1)(<br />
m+<br />
1)<br />
( −1) ( m −1)!<br />
=<br />
=<br />
m+ 1 m 1 m 1 m 1 m m 1 m 1<br />
( 2m+ 1) 2<br />
⎛ + ⎞<br />
!<br />
⎛ + ⎞ ⎛ − ⎞<br />
! 2<br />
⎛ + ⎞<br />
!<br />
⎛ − ⎞<br />
!<br />
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠<br />
m = odd
<strong>21</strong>.2 <strong>Legendre</strong> Functions of the Second Kind (Second Solution)<br />
<strong>21</strong>.2.1 Second Kind:<br />
1.<br />
2.<br />
3.<br />
4.<br />
<strong>21</strong>.2.2 Recursions<br />
Q n(t) satisfies all the recurrence relations of P n(t).<br />
<strong>21</strong>.2.3 Property<br />
<strong>21</strong>.2.4 Newman Formula<br />
<strong>21</strong>.3 Associated <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.3.1 Definition<br />
If m is a positive integer and –1 ≤ t ≤ 1, then<br />
m<br />
where P () t is known as the associated <strong>Legendre</strong> function or Ferrers’ functions.<br />
<strong>21</strong>.3.2 Rodrigues Formula<br />
<strong>21</strong>.3.3 Properties<br />
1.<br />
Q<br />
0<br />
© 1999 by CRC Press LLC<br />
1 1+<br />
t<br />
= ln , t < 1;<br />
2 1−<br />
t<br />
1 1 + t<br />
Q1() t = tln<br />
− 1, t < 1;<br />
2 1 − t<br />
2n+ 1 n<br />
Qn+ 1() t = tQn() t − Qn−1(), t n = 12 , ,L<br />
n + 1 n + 1<br />
[ 1(<br />
n−1)]<br />
∑<br />
2<br />
2n−4k−1 Qn() t = Pn() t Q0() t −<br />
Pn−2k−1(), t t < 1, n=<br />
12 , , L<br />
( 2k+ 1)(<br />
n−k) k=<br />
0<br />
for [ ( n −1)]<br />
1<br />
see <strong>21</strong>.1.1.<br />
2<br />
n<br />
−<br />
P () t = ( −1)<br />
m m<br />
n<br />
m ( 1 − t )<br />
Pn() t = n<br />
2 n!<br />
( n−m)! m<br />
Pn() t<br />
( n+ m)! 1<br />
Qn() t =<br />
2<br />
∞<br />
∑<br />
1<br />
= ( 2n+ 1)<br />
Pn( t) Qn( x)<br />
x − t<br />
∫<br />
1<br />
−1<br />
n=<br />
0<br />
Pn( x)<br />
dx, n = 012 , , L<br />
t − x<br />
m<br />
m 2 m/<br />
2 d Pn() t<br />
Pn() t = ( 1− t )<br />
m = 12 , , L,<br />
n<br />
m<br />
dt<br />
2 m / 2<br />
d<br />
dt<br />
n+ m<br />
n+ m<br />
2 n<br />
( t − 1) , m = 12 , , L,<br />
n; n+ m ≥ 0
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
0<br />
P () t = P() t<br />
n n<br />
m<br />
m<br />
m<br />
( n− m+ 1) P () t − ( 2n+ 1) tP () t + ( n+ m) P () t = 0<br />
<strong>21</strong>.3.4 Differential Equation<br />
<strong>21</strong>.3.5 Schlafli Formula<br />
where C is any regular closed curve surrounding the point t and taking it counterclockwise.<br />
<strong>21</strong>.4 Bounds for <strong>Legendre</strong> <strong>Polynomials</strong><br />
<strong>21</strong>.4.1 Stieltjes Theorem<br />
<strong>21</strong>.4.2 Second Stieltjes Theorem<br />
© 1999 by CRC Press LLC<br />
n+<br />
1 n<br />
n−1<br />
2 1/ 2 m 1 m+<br />
1 m+<br />
1<br />
( 1 − t ) Pn( t)<br />
= [ Pn+ 1 ( t) − Pn−1( t)]<br />
2n+ 1<br />
2 1/ 2 m 1<br />
m−1<br />
m−1<br />
( 1 − t ) Pn( t)<br />
= [( n+ m)( n+ m−1) Pn−1( t) −( n− m+ 1)( n− m+ 2)<br />
Pn+ 1 ( t)]<br />
2n+ 1<br />
m<br />
2 −1/ 2 m<br />
m−1<br />
P () t = 2mt( 1−t ) P () t − [ n( n+ 1) −m( m−1)] P () t<br />
n<br />
n<br />
m+<br />
1 2 −1/<br />
2<br />
m<br />
m<br />
P ( t) = ( t −1) [( n−m) tP ( t) − ( n+ m) P ( t)]<br />
n<br />
m m<br />
2 1/ 2 m−1<br />
P () t = P () t + ( 2n+ 1)( t −1)<br />
P () t<br />
n+<br />
1 n−1<br />
P () t = ( t −1)<br />
m<br />
n<br />
2 m/<br />
2<br />
1<br />
−1<br />
m<br />
d Pn() t<br />
m<br />
dt<br />
m m<br />
P t P t dt = 0 k ≠ n<br />
∫ n () k ()<br />
1<br />
∫<br />
−1<br />
m 2 2(<br />
n+ m)!<br />
[ Pn( t)] dt =<br />
( 2n+ 1)(<br />
n−m)! n<br />
n<br />
n−1<br />
2 m<br />
m<br />
d P () () ⎡<br />
2 ⎤<br />
2 n t dPn t<br />
m m<br />
( 1−t) − 2t+ ( + 1)<br />
− () 0<br />
2<br />
⎢nn<br />
2 ⎥ Pn t =<br />
dt dt<br />
⎣⎢<br />
( 1−<br />
t )<br />
⎦⎥<br />
n<br />
m 2 m/<br />
2<br />
∫ n n+ m+<br />
1<br />
C<br />
( n m)!<br />
Pn() t<br />
t<br />
jn!<br />
( )<br />
+<br />
= 1 −<br />
2π<br />
n<br />
2<br />
( x − 1)<br />
2 ( x − t)<br />
Pn<br />
(cos )<br />
, , n , ,<br />
n sin γ<br />
4 1<br />
≤ 2 0 < γ < π = 12L<br />
π γ<br />
P () t − P () t <<br />
n n+2<br />
4<br />
π<br />
n + 2<br />
dx
<strong>21</strong>.4.3<br />
<strong>21</strong>.4.4<br />
<strong>21</strong>.5 Table of <strong>Legendre</strong> and Associate <strong>Legendre</strong> Functions<br />
TABLE <strong>21</strong>.2 Properties of <strong>Legendre</strong> and Associate <strong>Legendre</strong> Functions [P n(t) = <strong>Legendre</strong> Functions,<br />
= Associate <strong>Legendre</strong> Functions, Q n(t) = <strong>Legendre</strong> Functions of the Second Kind]<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
14.<br />
15.<br />
16.<br />
17.<br />
18.<br />
19.<br />
20.<br />
<strong>21</strong>.<br />
22.<br />
dPn t<br />
dt<br />
()<br />
<<br />
© 1999 by CRC Press LLC<br />
2 n<br />
π 1 − t<br />
2<br />
, t < 1, n = 12 , , L<br />
6 2 1 1<br />
Pn+ 1()<br />
t + Pn() t <<br />
,<br />
π n 1 − t<br />
1<br />
1− 2tx+<br />
x<br />
P () t =<br />
n<br />
P ()= t 1<br />
0<br />
P<br />
2n<br />
P n+<br />
2<br />
∞<br />
∑ n<br />
n=<br />
0<br />
n<br />
= P () t x t ≤ 1 x < 1<br />
[ n / 2] k n 2k<br />
∑<br />
k=<br />
0<br />
−<br />
( −1) ( 2n−2k)! t<br />
n<br />
2 k!( n−k)!( n−2k)! n<br />
1 ( 1) ( 2n)!<br />
( 0)<br />
= 2<br />
2n2 n 2 (!) n<br />
− ⎛ ⎞<br />
⎜ ⎟ =<br />
⎝ ⎠<br />
−<br />
2 1( 0) = 0<br />
P ( − t) = P ( t) P ( − t) = −P<br />
( t)<br />
2n 2n 2n+ 1 2n+ 1<br />
n<br />
P ( − t) = ( −1 ) P ( t)<br />
n<br />
n<br />
P() 1 = 1 n= 012 ,, L ; P(<br />
− 1) = ( −1)<br />
n n<br />
n<br />
1 d 2 n<br />
Pn() t = ( t − 1)<br />
= Rodrigues formula, n n<br />
2 n!<br />
dt<br />
( n+ 1) Pn+ 1() t − ( 2n+ 1) tPn() t + nPn−1 () t = 0<br />
P′ () t − 2tP′ () t + P′ () t − P () t = 0<br />
n+ 1 n n−1 n<br />
P′ () t = P () t + 2tP′<br />
() t − P′ () t<br />
n− 1 n n n+<br />
1<br />
P′ () t = P () t + 2tP′<br />
() t − P′ () t<br />
n+ 1 n n n−1<br />
P′ () t − tP′ () t = ( n+ 1)<br />
P () t<br />
n+ 1 n n<br />
tP′ () t − P′ () t = nP() t<br />
n n−1 n<br />
Pn′ + 1() t − Pn′ −1()<br />
t = ( 2n+ 1)<br />
Pn () t<br />
2<br />
( 1 − t ) P′ () t = nP () t −ntP<br />
() t<br />
n n−1 n<br />
P () t < 1, − 1< t < 1<br />
n<br />
∑<br />
n k<br />
n<br />
( 1)<br />
( 1) ( 2n 2k 1)!<br />
P2n() t = t<br />
2n1 2 ( 2k)!( n k 1)!(<br />
n k)!<br />
− − + −<br />
−<br />
+ − −<br />
k=<br />
0<br />
2<br />
( 1− t ) Pn′ ( t) = ( n+ 1)[<br />
tPn ( t) − Pn + ( t)]<br />
1<br />
∫<br />
−1<br />
Pn t dt ()<br />
= 0<br />
1<br />
t < 1<br />
n<br />
[ n / 2]<br />
= n = even ; [ n/ 2] = ( n− 1)/ 2 n = odd<br />
2<br />
n<br />
2k<br />
n = 12 , ,L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
n = 12 , ,L<br />
n = 12 , ,L<br />
n = 12 , ,L<br />
n = 12 , ,L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
n = 12 , ,L<br />
Pn ()≤ t 1 t ≤ 1<br />
m<br />
Pn() t
TABLE <strong>21</strong>.2 Properties of <strong>Legendre</strong> and Associate <strong>Legendre</strong> Functions [P n(t) = <strong>Legendre</strong> Functions,<br />
= Associate <strong>Legendre</strong> Functions, Q n(t) = <strong>Legendre</strong> Functions of the Second Kind] (continued)<br />
23.<br />
24.<br />
25.<br />
26.<br />
27.<br />
28.<br />
29.<br />
30.<br />
31.<br />
32.<br />
33.<br />
34.<br />
35.<br />
36.<br />
37.<br />
38.<br />
39.<br />
40.<br />
41.<br />
∫<br />
−1<br />
© 1999 by CRC Press LLC<br />
1<br />
1<br />
∫<br />
−1<br />
1<br />
2<br />
1<br />
2<br />
1<br />
∫<br />
−1<br />
Pn() t Pm() t dt = 0<br />
n≠m 2 2<br />
[ Pn ( t)] dt =<br />
2n+ 1<br />
1<br />
∫<br />
−1<br />
1<br />
∫<br />
−1<br />
m<br />
mm−2 m− s+<br />
2<br />
t Ps t dt =<br />
m+ s+ 1 m+ s− 1 m+<br />
1<br />
()<br />
( ) L(<br />
)<br />
( )( ) L(<br />
)<br />
m<br />
m−1 m−3 m− s+<br />
2<br />
t Ps t dt =<br />
m+ s+ 1 m+ s− 1 m+<br />
2<br />
()<br />
( )( ) L(<br />
)<br />
( )( ) L(<br />
)<br />
2n<br />
tPn() t Pn−1() t dt=<br />
2<br />
4n−1 1<br />
P ′ ∫ n() t Pn+ 1()<br />
t dt =<br />
−1<br />
1<br />
2n<br />
tP′ = ∫ n() t Pn() t dt<br />
2n+ 1<br />
−1<br />
1<br />
−1<br />
2<br />
n = 012 ,, L<br />
ms , = even<br />
ms , = odd<br />
n = 12 , ,L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
2<br />
( 1− t ) P′ () t P′ () t dt = 0<br />
∫ n k k ≠ n<br />
1<br />
−1/<br />
2<br />
( 1 − t) P ( ) = ∫ n t dt<br />
−1<br />
1<br />
∫<br />
−1<br />
2 2<br />
2n+ 1<br />
2<br />
2nn ( + 1)<br />
t Pn+ 1() t Pn−1() t dt = 2<br />
( 4n − 1)( 2n+ 3)<br />
1<br />
2<br />
( t − 1)<br />
P ∫ n+ 1()<br />
t Pn ′ () t dt =<br />
−1<br />
1<br />
∫<br />
−1<br />
1<br />
∫<br />
−1<br />
n+<br />
1 2<br />
n 2 (!) n<br />
t Pn() t dt =<br />
( 2n+ 1)!<br />
2nn ( + 1)<br />
( 2n+ 1)( 2n+ 3)<br />
2 2<br />
2 2 2 ⎡(<br />
n + 1)<br />
n ⎤<br />
t [ Pn( t)] dt = ⎢ +<br />
2<br />
⎥<br />
( 2n+ 1)<br />
⎣ 2n+ 3 2n−1⎦ m<br />
m 2 m/<br />
2 d<br />
Pn() t = ( 1 −t<br />
) P t m n ()<br />
dt<br />
m<br />
Pn() t<br />
t<br />
n<br />
n!<br />
n m<br />
m d<br />
n m<br />
dt<br />
t<br />
( ) [( ) ]<br />
1 2<br />
= 1− 2<br />
+<br />
/ 2<br />
+<br />
2<br />
−1<br />
n<br />
−m<br />
m ( n−m)! m<br />
Pn() t = ( −1)<br />
Pn() t<br />
( n+ m)! 0<br />
P () t = P(), t<br />
m<br />
P () t = 0 for m> n<br />
n n n<br />
m<br />
m<br />
m<br />
( n− m+ 1) P () t − ( 2n+ 1) tP () t + ( n+ m) P () t = 0<br />
n+<br />
1 n<br />
n−1<br />
2 1/ 2 m 1 m+<br />
1 m+<br />
1<br />
( 1−<br />
t ) Pn( t)<br />
= [ Pn+ 1 ( t) − Pn−1( t)]<br />
2n+ 1<br />
n = 012 ,, L<br />
n = 12 , ,L<br />
n = 12 , ,L<br />
n = 012 ,, L<br />
n = 012 ,, L<br />
m > 0<br />
m+ n ≥ 0<br />
m<br />
Pn() t
TABLE <strong>21</strong>.2 Properties of <strong>Legendre</strong> and Associate <strong>Legendre</strong> Functions [P n(t) = <strong>Legendre</strong> Functions,<br />
= Associate <strong>Legendre</strong> Functions, Q n(t) = <strong>Legendre</strong> Functions of the Second Kind] (continued)<br />
42.<br />
43.<br />
44.<br />
45.<br />
46.<br />
47.<br />
48.<br />
49.<br />
50.<br />
51.<br />
52.<br />
53.<br />
54.<br />
55.<br />
56.<br />
57.<br />
2 1/ 2 m 1<br />
m−1<br />
( 1 − t ) Pn( t)<br />
= [( n+ m)( n+ m−1) Pn−1( t)<br />
2n+ 1<br />
m−1<br />
−( n− m+ 2)<br />
P ( t)]<br />
© 1999 by CRC Press LLC<br />
n+<br />
1<br />
m+<br />
1 2 −1/ 2 m<br />
m−1<br />
P () t = 2mt( 1−t) P () t − [ n( n + 1) −m( m −1)]<br />
P () t<br />
1<br />
∫<br />
n<br />
−1<br />
1<br />
∫<br />
−1<br />
n<br />
m m<br />
P () t P () t dt = 0<br />
k ≠ n<br />
n<br />
k<br />
m 2 2 ( n+ m)!<br />
[ Pn( t)] dt =<br />
2n+ 1 ( n−m)! m n+ m m<br />
P ( − t) = ( −1)<br />
P ( t)<br />
n<br />
P n<br />
n<br />
m<br />
( ± 1) = 0 m > 0<br />
n<br />
1<br />
1 ( − 1) ( 2n+ 1)!<br />
P2n( 0) = 0 P2n+<br />
1(<br />
0)<br />
= 2n2 2 (!) n<br />
P n<br />
P<br />
m<br />
( 0) = 0<br />
n+ m = odd<br />
( 0) = ( −1)<br />
m ( n−m)/ 2<br />
n<br />
1<br />
∫<br />
−1<br />
m k<br />
Pnt Pnt 2<br />
− t<br />
−1<br />
dt =<br />
( n+ m)!<br />
n<br />
2 [( n− m) / 2]![( n+ m)<br />
/ 2]!<br />
() ()( 1 ) 0<br />
k ≠ m<br />
1<br />
2 −1/<br />
2<br />
( 1 − t ) P = ∫ 2m<br />
( t) dt<br />
−1<br />
1<br />
2 −1<br />
2<br />
t( 1 − t ) P ∫ 2m+ 1(<br />
t) dt =<br />
−1<br />
1<br />
∫<br />
t<br />
1<br />
0<br />
1 ⎡ Γ(<br />
2 + m)<br />
⎤<br />
⎢ ⎥<br />
⎣ m!<br />
⎦<br />
1<br />
3<br />
/ Γ( 2 + m) Γ(<br />
2 + m)<br />
m!( m+<br />
1)!<br />
1<br />
Pn () t dt = [ Pn− 1() t − Pn+ 1()]<br />
t<br />
2n+ 1<br />
n k<br />
( − 1) Γ ( n+ k + 1)<br />
q<br />
t P t dt = q+<br />
1 ∫ n () Γ(<br />
) ∑ k<br />
2 k! Γ ( n− k + 1)( q+ k + 2)<br />
1<br />
1 2<br />
∫<br />
0<br />
− /<br />
1<br />
∫<br />
0<br />
Q<br />
0<br />
k=<br />
0<br />
⎧ n / 2<br />
⎪2(<br />
−1)<br />
⎪ 2n+ 1<br />
t Pn() t dt = ⎨<br />
⎪<br />
⎪2(<br />
−1)<br />
⎩<br />
⎪<br />
2n+ 1<br />
1/ 2<br />
t Pn() t dt<br />
1 1+<br />
t<br />
= ln ,<br />
2 1−<br />
t<br />
( n−1)/<br />
2<br />
⎧<br />
( n+<br />
2)/ 2<br />
⎪ 2( −1)<br />
⎪<br />
⎪<br />
( 2n− 1)( 2n+ 3)<br />
= ⎨<br />
⎪<br />
( n+<br />
3)/ 2<br />
⎪ 2( −1)<br />
⎪<br />
⎩(<br />
2n− 1)( 2n+ 3)<br />
t < 1<br />
2<br />
n = even<br />
n = odd<br />
n = even<br />
n = odd<br />
n<br />
n+ m = even<br />
q >−1<br />
m<br />
Pn() t
TABLE <strong>21</strong>.2 Properties of <strong>Legendre</strong> and Associate <strong>Legendre</strong> Functions [P n(t) = <strong>Legendre</strong> Functions,<br />
= Associate <strong>Legendre</strong> Functions, Q n(t) = <strong>Legendre</strong> Functions of the Second Kind] (continued)<br />
58.<br />
59.<br />
60.<br />
61.<br />
62.<br />
63.<br />
64.<br />
65.<br />
References<br />
1 1+<br />
t<br />
Q1() t = tln<br />
− 1= tQ0() t − 1, t < 1<br />
2 1−<br />
t<br />
2n+ 1 n<br />
Qn+ 1() t = tQn() t − Qn−1(), t n = 12 , ,L<br />
n + 1 n + 1<br />
Q () t = P() t Q () t −<br />
n n<br />
Abramowitz, M. and I. S. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New<br />
York, NY, 1965.<br />
Andrews, L. C., Special Functions for Engineers and Applied Mathematicians, MacMillan Publishing<br />
Co. New York, NY. 1985.<br />
Hochstadt, H., The Functions of Mathematical Physics, Dover Publications Inc., New York, NY, 1986.<br />
McLachlan, N. W., Bessel Functions for Engineers, 2nd Edition, Oxford University Press, London, 1961.<br />
Sansone, G., Orthogonal Functions, Interscience Publishers, New York, NY., 1959.<br />
© 1999 by CRC Press LLC<br />
0<br />
[ 1 ( n−1)]<br />
2<br />
∑<br />
k=<br />
0<br />
1 1 Pn( x)<br />
Qn() t =<br />
dx<br />
2 ∫−1<br />
t−x n = 012 ,, L<br />
Q′ () t − 2tQ′ () t + Q′ () t − Q () t = 0<br />
n+ 1 n n−1 n<br />
Qn′ + 1() t − tQn′ () t − ( n+ 1) Qn() t = 0<br />
Qn′ + 1() t − Qn′ −1()<br />
t = ( 2n+ 1)<br />
Qn() t<br />
2n−4k−1 Pn−2k−1(), t t < 1<br />
( 2k+ 1)(<br />
n−k) n+<br />
1<br />
Q ( − t) = −Q ( t), Q ( − t) = ( − 1) Q ( t), n=<br />
12 , , L<br />
0 0<br />
n<br />
n<br />
m<br />
Pn() t