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21—30 of 706 matching pages

21: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►with

a_{n}\neq 0

\forall n

, can be computed recursively for

n=2,3,\dots

. Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. … ►In the notation of §3.6(v) we have

M=10

and

\epsilon=\tfrac{1}{2}\times 10^{-8}

. … ►For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).

22: 5.11 Asymptotic Expansions

… ►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). …

23: 24.2 Definitions and Generating Functions

… ►

Table 24.2.3: Bernoulli numbers

B_{n}=N/D

► ►►►

$n$	$N$	$D$	$B_{n}$
…
$8$	$-1$	$30$	$-3.33333\;3333$	$\times 10^{-2}$
…

► ►

Table 24.2.4: Euler numbers

E_{n}

► ►►►

$n$	$E_{n}$
…
$8$	$1385$
…

► ►

Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

► ►►►►

	$k$
…
$14$	$\tfrac{7}{6}$	$0$	$-\tfrac{691}{30}$	$0$	$\tfrac{455}{6}$	$0$	$-\tfrac{1001}{10}$	$0$	$\tfrac{143}{2}$	$0$	$-\tfrac{1001}{30}$	$0$	$\tfrac{91}{6}$	$-7$	$1$
$15$	$0$	$\tfrac{35}{2}$	$0$	$-\tfrac{691}{6}$	$0$	$\tfrac{455}{2}$	$0$	$-\tfrac{429}{2}$	$0$	$\tfrac{715}{6}$	$0$	$-\tfrac{91}{2}$	$0$	$\tfrac{35}{2}$	$-\tfrac{15}{2}$	$1$

► ►

Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

► ►►►

	$k$
…
$14$	$0$	$-38227$	$0$	$62881$	$0$	$-31031$	$0$	$7293$	$0$	$-1001$	$0$	$91$	$0$	$-7$	$1$
…

►

24: 23.21 Physical Applications

… ►For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). … ►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation …

25: Bibliography N

… ►

G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

… ►

G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

ⓘ

External Links:: ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt
Referenced by:: §11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).
Encodings:: BibTeX

… ►

E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

ⓘ

Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

…

26: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82). … ►The implicit boundary conditions taken here are that the

\phi_{n}(x)

and

\phi_{n}^{\prime}(x)

vanish as

x\to\pm\infty

, which in this case is equivalent to requiring

\phi_{n}(x)\in L^{2}\left(X\right)

, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►More generally, for

f\in C(X)

x\in X

, see (1.4.24), … ►See Titchmarsh (1962a, pp. 87–90) for a first principles derivation for the case

\Re\nu\geq 1

. … ►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). …

27: Bibliography V

… ►

J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

… ►

B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

ⓘ

External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

… ►

L. Vietoris (1983) Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

ⓘ

External Links:: ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt
Referenced by:: §8.10.
Encodings:: BibTeX

… ►

A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 1(1), pp. 58–59.
External Links:: MathReview, ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

…

28: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …(

\nu\left(1\right)

is defined to be 0.) …It can be expressed as a sum over all primes

p\leq x

: … ►is the sum of the

\alpha

th powers of the divisors of

n

, where the exponent

\alpha

can be real or complex. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
…

►

29: 3.9 Acceleration of Convergence

… ►It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. It may even fail altogether by not being limit-preserving. … ►Let

k

be a fixed positive integer. … ►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: … ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). …

30: 20.5 Infinite Products and Related Results

…