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1: 19.8 Quadratic Transformations

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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►As $n\to \mathrm{\infty }$, $a_n$ and $g_n$ converge to a common limit $M(a_0,g_0)$ called the AGM (Arithmetic-Geometric Mean) of $a_0$ and $g_0$. …showing that the convergence of $c_n$ to 0 and of $a_n$ and $g_n$ to $M(a_0,g_0)$ is quadratic in each case. … ►

19.8.5 $$K\left(k\right)=\frac{\pi }{2M(1,k^{\prime })},$$ .

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

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http://dlmf.nist.gov/19.8.E5

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Annotations for §19.8(i), §19.8 and Ch.19

… ►Again, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. …

2: 1.15 Summability Methods

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Abel Summability

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Abel Means

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and … ►If $f(\theta )$ is periodic and integrable on $[0,2\pi ]$, then as $n\to \mathrm{\infty }$ the Abel means $A(r,\theta )$ and the (C,1) means ${\sigma }_n(\theta )$ converge to … ►

Abel Summability

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3: 22.18 Mathematical Applications

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). …

4: 1.2 Elementary Algebra

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§1.2(iv) Means

►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is … ►The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_1,a_2,\mathrm{\dots },a_n$ are given by … ►If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with … ►

$M(1)=A,$

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5: 2.10 Sums and Sequences

… ►Another version is the Abel–Plana formula: … ►

The first infinite integral in (2.10.2) converges.

… ►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula …

6: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 46(1991), no. 4, pp. 1–45

External Links:

ISSN 0042-1316, MathReview (P. E. Markov), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

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7: 20 Theta Functions

Chapter 20 Theta Functions

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8: 1.7 Inequalities

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§1.7(iii) Means

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1.7.7 $$H\le G\le A,$$

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Symbols:

$A$: arithmetic mean, $G$: geometric mean and $H$: harmonic mean

A&S Ref:

3.2.1

Permalink:

http://dlmf.nist.gov/1.7.E7

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Annotations for §1.7(iii), §1.7 and Ch.1

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1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

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Annotations for §1.7(iii), §1.7 and Ch.1

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1.7.9 $$M(r)\le M(s),$$ ,

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Symbols:

$M(r)$: weighted mean

A&S Ref:

3.2.4

Permalink:

http://dlmf.nist.gov/1.7.E9

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Annotations for §1.7(iii), §1.7 and Ch.1

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9: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

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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:

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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).

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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:

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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:

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10: Bibliography G

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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).

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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).

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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:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

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