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j
but standardized as in A.49.


Expansion of Complex Exponential



m m
(A.47)
exp( i π ω x) = fj Cj (x)
j=0


m
2
m j
(A.48)
fj = i “(m) (j + m) Jj+m ( π ω )
πω
Standardization:

“(n + 2m)
(m)
(A.49)
Cn (1) =
n! “(2m)
A.5. GEGENBAUER POLYNOMIALS 503

Asymptotics of Endpoint Values

n
1 27
∼ n’∞
n
(A.50)
Cn (1) , [m = n]
3πn 4

2m’1/2
1 1

∼ (n + 1) exp {2 m [1 ’ log(2) ] } ,
2m’1
m
Cn (1)
m
π
¬xed m, n ’ ∞ (A.51)
Inequality:

|Cn (x)| ¤ Cn (1) for m > 1/2, x ∈ [’1, 1]
(m) (m)
(A.52)
Uniformity Ratio

m
C2n (1) “(2n + 2m) n! “(m)
ρ2n (m) ≡ m (A.53)
=
|C2n (0)| (2n)! “(2m) “(n + m)

2n
C2n (1) “(6n) n!
ρ2n (2n) ≡ 2n (A.54)
=
|C2n (0)| 2n “(4n) “(3n)
Three-Term Recurrence:

(m) (m)
≡1 (x) ≡ 2 m x (A.55)
C0 ; C1
(m) (m) (m)
(n + 1) Cn+1 (x) = 2 (n + m) x Cn (x) ’ (n + 2m ’ 1) Cn’1 (x) (A.56)
Note: the recurrence relation is weakly unstable for m > 0, and the instability increases
as m increases. Consequently, library software usually calculates the spherical harmonics
using a different, more complicated recurrence relation (see book by Belousov, 1962). How-
ever, our own numerical experiments, perturbing the recurrence relation by ρ(n) where
ρ(n) is a different random number between 0 and 1 for each m, have shown the maximum
error for all x and all polynomial degrees < 100 is only 263 for m = 24.5 and 1247 for
m = 49.5. Nehrkorn (1990) found the same through his more detailed experiments. This
suggests that the recurrence relation is actually quite satisfactory if a relative error as large
as 1.E-10 is acceptable.

Differentiation Rule:

d (m) (m+1)
(A.57)
Cn (x) = 2 m Cn’1 (x)
dx
Phillips and Karageorghis (1990) derive formulas for the coef¬cients of integrated ex-
pansions of Gegenbauer polynomials.

Collocations Points (Gaussian Quadrature Abscissas):

See Davis & Rabinowitz, Methods of Numerical Integration. Canuto et al. (1988) give
a short FORTRAN subroutine to compute the Lobatto grid for general order in their Ap-
pendix C.
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
504

Special Cases:

n (0)
[Chebyshev] (A.58)
Tn (x) = Cn (x)
2
(1/2)
[Legendre] (A.59)
Pn (x) = Cn (x)
(1)
[Chebyshev of the 2d Kind] (A.60)
Un (x) = Cn (x)

Gegenbauer Polynomials: Explicit Form


m m
C0 = 1; C1 (x) = 2m x
2m(1 + m) x2 ’ m
m
C2 =
8 4
m + 4m2 + m3 x3 ’ 2m(1 + m)x
m
C3 =
3 3
24 22 1
m + 4m3 + m2 + 4m x4 ’ (2m3 + 6m2 + 4m)x2 + m(1 + m)
m
C4 =
3 3 2
4 5 8 4 28 3 40 2 32
x5
m
C5 = m+ m+ m+ m+
15 3 3 3 5
44 44
’ m + 8m3 + m2 + 8m x3 + (m3 + 3m2 + 2m)x (A.61)
3 3

Endpoint values

m m
C0 (1) = 1; C1 (1) = 2m
43 2
= m + 2m2 ; m + 2m2 + m
m m
C2 (1) C3 (1) =
3 3
24 11 1
m + 2m3 + m2 + m
m
C4 (1) =
3 6 2
45 44 73 52 2
m
(A.62)
C5 (1) = m+ m+ m+ m+ m
15 3 3 3 5
A.6. HERMITE POLYNOMIALS: HN (X) 505

A.6 Hermite Polynomials: Hn (x)


f (x) ’ 0 |x| ’ ∞
Applications: doubly in¬nite interval; as



Basis Functions:


φn (x) ≡ exp[’x2 /2] Hn (x) (A.63)

Interval

x ∈ [’∞, ∞] (A.64)




Weight:


ρ(x) ≡ exp[’x2 ] (A.65)

Inner Product:

∞ √
e’x Hm (x) Hn (x) dx = δmn
2
π 2n n! (A.66)
’∞

Standardization:

Hn (x) = 2n xn + · · · (A.67)

Three-Term Recurrence and Starting Values:


H0 ≡ 1 H1 (x) ≡ 2 x (A.68)
;

Hn+1 (x) = 2 x Hn (x) ’ 2 n Hn’1 (x) (A.69)
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
506

Differentiation Rules:

dHn
≡ 2 n Hn’1 (x) all n (A.70)
dx
Collocation Points (Gaussian Abscissas): Abramowitz & Stegun(1965), NBS Handbook, pg.
924 , Davis and Rabinowitz, Numerical Integration.


ψn (x)2 dx = 1 for all n may be com-
The orthonormal Hermite functions ψn such that ’∞
puted by

ψ0 ≡ π ’1/4 exp(’(1/2)x2 ) ψ1 (x) ≡ (A.71)
; 2 x ψ0

x ψn (x) ’
2 n
(A.72)
ψn+1 (x) = ψn’1 (x)
n+1 n+1


and the derivatives ψn,x from

ψ0,x ≡ ’xψ0 ψ1,x (x) ≡ ’x ψ1 + (A.73)
; 2x ψ0

= ’ x ψn+1 + 2 n ψn (A.74)
ψn+1,x

The orthonormal Hermite functions are better-conditioned than their unnormalized coun-
terparts.

Inequality:


|ψn (x)| =¤ 0.816 ∀ x (A.75)



Expansion of a Gaussian:


2
(A.76)
exp(’x ) = a2m ψ2m (x)
m=0



π 1/4
2/3 2m!
(’1)m (A.77)
a2 =
6m m!
1
∼ m’∞
(’1)m 0.9709835 m (A.78)
,
3 (2m)1/4

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