A.7 Rational Chebyshev Functions on an In¬nite Interval:

T Bn (y)

Applications: Doubly in¬nite interval

y ∈ [’∞, ∞]

Interval:

Let L denote a constant map parameter, user-chosen to match width of the function to

be expanded. (Some experimentation may be necessary.) Then

Chebyshev de¬nition:

y

(A.79)

T Bn (y) = Tn

L2 + y 2

Trigonometric de¬nition:

y

cos n arccot (A.80)

T Bn (y) =

L

(A.81)

= cos(nt)

where

y

←’ t = arccot (A.82)

y = L cot(t)

L

Weight:

L

ρ(y) ≡ (A.83)

L2 + y 2

Inner Product:

±

π m=n=0

∞

L

(A.84)

0 m=n

T Bm (y) T Bn (y) dy =

L2 + y2

’∞

π/2 m=n>1

Asymptotic Behavior:

T Bn (y) ∼ 1 as |y| ’ ∞ for all n (A.85)

(Note that this basis set may usually be applied without modi¬cation to compute

solutions that tend to 0 as |y| ’ ∞.)

Derivatives:

Apply the trigonometric form (A.81) in combination with Table E“4.

References: Boyd (1987a).

APPENDIX A. A BESTIARY OF BASIS FUNCTIONS

508

A.8 Laguerre Polynomials: Ln (x)

f (x) ’ 0 as x ’ ∞

Applications: Semi-In¬nite interval;

Laguerre functions (the basis set):

φn (x) ≡ exp[’x/2] Ln (x) (A.86)

x ∈ [0, ∞]

Interval:

Weight:

ρ(x) ≡ exp[’x/2] (A.87)

Inner Product:

∞

e’x Lm (x) Ln (x) dx = δmn (A.88)

0

Standardization:

(’1)n n

x + ··· (A.89)

Ln (x) =

n!

Inequality:

|φn (x)| = |exp(’x/2) Ln (x)| ¤ 1 (A.90)

Three-Term Recurrence and Starting Values:

L1 (x) = ’x + 1 (A.91)

L0 (x) = 1 ;

(n + 1) Ln+1 (x) = {(2 n + 1) ’ x} Ln (x) ’ n Ln’1 (x) (A.92)

Differential Relations:

dm

(x) = (’1)m L(m) (x) (A.93)

L

m n+m n

dx

(m)

where the Ln (x) are the generalized Laguerre polynomials. These can be computed by

using the recurrence

(m) (m)

L0 (x) ≡ 1 L1 (x) ≡ 1 + m ’ x (A.94)

;

(m) (m) (m)

(n + 1) Ln+1 (x) = {(2n + m + 1) ’ x} Ln (x) ’ (n + m) Ln’1 (x) (A.95)

Collocation Points:

Abramowitz & Stegun(1965), NBS Handbook, pg. 923.

A.9. RATIONAL CHEBYSHEV FUNCTIONS: T LN (Y ) 509

A.9 Rational Chebyshev Functions on Semi-In¬nite Inter-

val: T Ln (y)

y ∈ [0, ∞]

Interval:

(Generalization: r ∈ [r0 , ∞] by substituting y = r ’ r0 in all formulas below.) Let L de-

note a constant map parameter, user-chosen to match width of the function to be expanded.

(Some experimentation may be necessary.) Then

Chebyshev de¬nition:

y’L

T Ln (y) ≡ Tn (A.96)

y+L

Trigonometric de¬nition:

y

T Ln (y) ≡ cos 2 n arccot (A.97)

L

≡ cos(nt) (A.98)

where

t y

y ≡ L cot2 ←’ t = 2 arccot (A.99)

2 L

Weight function:

1 L

ρ(y) ≡ (A.100)

y+L y

±

π m=n=0

∞

1 L

(A.101)

0 m=n

T Lm (y; L) T Ln (y; L) dy =

y+L y

0

π/2 m=n>0

Derivatives:

Apply trigonometric form (A.98) in Table E“6.

References: Boyd (1987b).

APPENDIX A. A BESTIARY OF BASIS FUNCTIONS

510

Table A.1: Flow Chart on the Choice of Basis Functions

If Basis Set is

f (x) is periodic Fourier series

f (x) is periodic & symmetric about x = 0 Fourier cosine

f (x) is periodic & antisymmetric about x = 0 Fourier sine

Chebyshev polys.

x ∈ [a, b] & f (x) is non-periodic

Legendre polys.

T Ln (y)

y ∈ [0, ∞] & f (y) decays exponentially as y ’ ∞

Laguerre functions

f (y) has asymptotic series

y ∈ [0, ∞] & T Ln (y) only

in inverse powers of y

T Bn (y)

y ∈ [’∞, ∞] & f (y) decays exponentially as y ’ ∞ sinc functions or

Hermite functions

f (y) has asymptotic series

y ∈ [’∞, ∞] & T Bn (y) only

in inverse powers of y

f (», θ) is a function of latitude and longitude spherical harmonics

A.10. GRAPHS OF CONVERGENCE DOMAINS IN THE COMPLEX PLANE 511

A.10 Graphs of Convergence Domains in the Complex Plane

Regions of convergence of various basis sets in the complex plane along with the expansion

interval appropriate to that basis. The cross-hatching indicates the region within which the

series converges. If f (x) denotes the function being expanded, then there is usually a pole

or branch point of f (x) on the curve bounding the region of convergence. In all cases,

the function f (x) has no singularities inside the convergence region; the size of the region

of convergence is thus controlled by the poles and branch points of the function which is

being approximated.

The shapes of the curves that bound that convergence regions, however, are determined

entirely by the basis set. Thus, a Chebyshev series always converges within a certain ellipse

while diverging outside that ellipse, but whether the ellipse is large or small depends on

the singularities of f (x), and which one is closest to the interval x ∈ [’1, 1].

FOURIER

Im(x)

Re(x)

Figure A.1: The convergence boundary for a Fourier series is the pair of straight lines,

Im(x) = constant

CHEBYSHEV

Im(x)

LEGENDRE

GEGENBAUER

Re(x)

Figure A.2: Chebyshev, Legendre, and Gegenbauer polynomials of all orders: ellipse with

foci at x = ±1. The foci are marked by black disks.

APPENDIX A. A BESTIARY OF BASIS FUNCTIONS

512

HERMITE x ∈ [’ ∞, ∞ ]

SINC