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21: 19.36 Methods of Computation

… ►If (19.36.1) is used instead of its first five terms, then the factor

(3r)^{-1/6}

in Carlson (1995, (2.2)) is changed to

(3r)^{-1/8}

. ►For both

R_{D}

and

R_{J}

the factor

(r/4)^{-1/6}

in Carlson (1995, (2.18)) is changed to

(r/5)^{-1/8}

when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►Accurate values of

F\left(\phi,k\right)-E\left(\phi,k\right)

for

k^{2}

near 0 can be obtained from

R_{D}

by (19.2.6) and (19.25.13). … ►

E\left(\phi,k\right)

can be evaluated by using (19.25.7), and

R_{D}

by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of

R_{J}

R_{D}

. …

22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Assume that

\mathcal{D}(T)

is dense in

V

, i. … ►

u_{\lambda}\in\mathcal{D}(T)

, corresponding to distinct eigenvalues, are orthogonal: i. … ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that

\mathcal{D}(T)=\mathcal{D}({T}^{*})

, as

f(x)

and

g(x)

satisfy the same boundary conditions and thus define the same domains. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals

[\lambda_{a},\lambda_{b}]\in(0,\infty)

, often called bands, when

q(x)

is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►,

\mathcal{D}(T)\subset\mathcal{D}({T}^{*})

and

{T}^{*}v=Tv

for

v\in\mathcal{D}(T)

. …

23: 3.2 Linear Algebra

… ►Assume that

\mathbf{A}

can be factored as in (3.2.5), but without partial pivoting. … ►where

\rho(\mathbf{A}\mathbf{A}^{\rm T})

is the largest of the absolute values of the eigenvalues of the matrix

\mathbf{A}\mathbf{A}^{\rm T}

; see §3.2(iv). … ►is called the characteristic polynomial of

\mathbf{A}

and its zeros are the eigenvalues of

\mathbf{A}

. … ►has the same eigenvalues as

\mathbf{A}

. … ►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).

E. Hairer, S. P. Nørsett, and G. Wanner (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

ⓘ

Notes:: A reprinting with corrections appeared in 2000.
External Links:: ISBN 3-540-56670-8, MathReview, ZentralBlatt
Referenced by:: §3.7(v), §3.7(i).
Encodings:: BibTeX

… ►

B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

ⓘ

External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.16.
Encodings:: BibTeX

… ►

H. W. Hethcote (1970) Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

… ►

L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

ⓘ

External Links:: Document
Referenced by:: §33.7.
Encodings:: BibTeX

… ►

K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

…

25: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8	9	10
…
8	1	8	28	56	70	56	28	8	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8
…
1	1	2	3	4	5	6	7	8	9
…
8	1	9	45	165	495	1287	3003	6435	12870

► …

26: 31.2 Differential Equations

… ►

31.2.3

\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\left(\frac{A}{z}+\frac{B}{z-1}+% \frac{C}{z-a}+\frac{D}{z^{2}}+\frac{E}{(z-1)^{2}}+\frac{F}{(z-a)^{2}}\right)W,

A+B+C=0

ⓘ

Defines:: $W(z)$ : solution (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : complex parameter, $A$ , $B$ , $C$ , $D$ , $E$ and $F$
Permalink:: http://dlmf.nist.gov/31.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.2(ii), §31.2 and Ch.31

… ►

$F$ -Homotopic Transformations

… ►By composing these three steps, there result

2^{3}=8

possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … ►There are

4!=24

homographies

\tilde{z}(z)=(Az+B)/(Cz+D)

that take

0,1,a,\infty

to some permutation of

0,1,a^{\prime},\infty

, where

a^{\prime}

may differ from

a

. … ►There are

8\cdot 24=192

automorphisms of equation (31.2.1) by compositions of

F

-homotopic and homographic transformations. …

27: 3.5 Quadrature

… ►About

2^{9}=512

function evaluations are needed. … ►For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). … ►

Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.

► ►►►

$\pm x_{k}$	$w_{k}$
0.07652 65211 33497 33375 5	0.15275 33871 30725 85069 8
…

► … ►

Table 3.5.20: Composite trapezoidal rule for the integral (3.5.45) with

\lambda=10

► ►►►

$h$	$\operatorname{erfc}\lambda$		$n$
…
$0.15$	$0.20884\;87588\;72946$	$\times 10^{-44}$	8
…

► … ►Table 3.5.21 supplies cubature rules, including weights

w_{j}

, for the disk

D

, given by

x^{2}+y^{2}\leq h^{2}

: …

28: 10.75 Tables

… ►

•

British Association for the Advancement of Science (1937) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $x=0(.001)16(.01)25$ , 10D; $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0.01(.01)25$ , 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ for small values of $x$ , as well as auxiliary functions to compute all four functions for large values of $x$ .

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Abramowitz and Stegun (1964, Chapter 9) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (10D for $n=0$ ), $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (8D for $n=0$ ), $J_{0}\left(j_{0,m}x\right)$ , $m=1(1)5$ , $x=0(.02)1$ , 5D. Also included are the first 5 zeros of the functions $xJ_{1}\left(x\right)-\lambda J_{0}\left(x\right)$ , $J_{1}\left(x\right)-\lambda xJ_{0}\left(x\right)$ , $J_{0}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{0}\left(x\right)J_{0}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{1}\left(\lambda x\right)-Y_{1}\left(x\right)J_{1}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{1}\left(x\right)J_{0}\left(% \lambda x\right)$ for various values of $\lambda$ and $\lambda^{-1}$ in the interval $[0,1]$ , 4–8D.

… ►

•

Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

… ►

•

Achenbach (1986) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0(.1)8$ , 19D or 19–21S.

…

29: Software Index

… ► ►►►►►►►

	Open Source						With Book			Commercial
…
8 Incomplete Gamma and Related Functions
…
9 Airy and Related Functions
…
11.16(v) $\mathbf{J}_{\nu}\left(z\right)$ , $\mathbf{E}_{\nu}\left(z\right)$ , $\mathbf{A}_{\nu}\left(z\right)$					a	✓			✓		✓	✓	a
…
19.39(iv) $R_{C}\left(x,y\right)$ , $R_{F}\left(x,y,z\right)$ , $R_{D}\left(x,y,z\right)$ , $R_{J}\left(x,y,z,p\right)$	✓	✓	✓	✓	a	✓		✓		✓				✓	Derive
…
34 3j, 6j, 9j Symbols
…

…

30: 1.12 Continued Fractions

… ►

C_{n}

is called the

n

th approximant or convergent to $C$ .

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►Define … ►A contraction of a continued fraction

C

is a continued fraction

C^{\prime}

whose convergents

\{C^{\prime}_{n}\}

form a subsequence of the convergents

\{C_{n}\}

C

. Conversely,

C

is called an extension of

C^{\prime}

. …