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