Abel means

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21: 9.17 Methods of Computation

… ►In the case of

\operatorname{Ai}\left(z\right)

, for example, this means that in the sectors

\tfrac{1}{3}\pi<|\operatorname{ph}z|<\pi

we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). …

22: Bibliography

… ►

S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.

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External Links:: MathReview
Referenced by:: §25.18(i).
Encodings:: BibTeX

►

G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,

\pi

, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:: ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:: For corrigendum see same journal v. 90 (1998), no. 2, p. 265
External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

…

23: 27.3 Multiplicative Properties

… ►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and …

24: 3.10 Continued Fractions

… ►For several special functions the

S

-fractions are known explicitly, but in any case the coefficients

a_{n}

can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … ►We continue by means of the rhombus rule … ►The

A_{n}

and

B_{n}

of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). …

25: 35.4 Partitions and Zonal Polynomials

… ►

Mean-Value

…

26: Errata

… ►

Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$ -argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$ , where $1\leq j,k\leq n$ . Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations. A reference to Ince (1926, §5.2) was added.

… ►

(10.9.26)

The factor on the right-hand side containing $\cos(\mu-\nu)\theta$ has been been replaced with $\cos\left((\mu-\nu)\theta\right)$ to clarify the meaning.

… ►

Equation (19.7.2)

The second and the fourth lines containing $k^{\prime}/ik$ have both been replaced with $-ik^{\prime}/k$ to clarify the meaning.

…

27: Bibliography G

… ►

L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

… ►

W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:: Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

… ►

W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:: Orthogonal polynomials and numerical methods
External Links:: ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt
Referenced by:: Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

…

28: 8.14 Integrals

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29: 22.10 Maclaurin Series

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30: Mathematical Introduction

… ► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$\mod$ or modulo	$m\equiv n\pmod{p}$ means $p$ divides $m-n$ , where $m$ , $n$ , and $p$ are positive integers with $m>n$ .
…

… ►This means that the variable

x

ranges from 0 to 1 in intervals of 0. …