Abel means
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1: 19.8 Quadratic Transformations
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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
… ►As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . …showing that the convergence of to 0 and of and to is quadratic in each case. … ►
19.8.5
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►Again, and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically.
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2: 1.15 Summability Methods
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Abel Summability
… ►Abel Means
… ► is a harmonic function in polar coordinates (1.9.27), and … ►If is periodic and integrable on , then as the Abel means and the (C,1) means converge to … ►Abel Summability
…3: 1.2 Elementary Algebra
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§1.2(iv) Means
►The arithmetic mean of numbers is … ►The geometric mean and harmonic mean of positive numbers are given by … ►If is a nonzero real number, then the weighted mean of nonnegative numbers , and positive numbers with … ►4: 22.18 Mathematical Applications
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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
… ►For any two points and on this curve, their sum , 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). …5: 2.10 Sums and Sequences
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►Another version is the Abel–Plana formula:
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(c)
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►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula
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The first infinite integral in (2.10.2) converges.
6: 1.7 Inequalities
7: 15.17 Mathematical Applications
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►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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8: 22.20 Methods of Computation
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§22.20(ii) Arithmetic-Geometric Mean
… ►Then as sequences , converge to a common limit , the arithmetic-geometric mean of . … ►The rate of convergence is similar to that for the arithmetic-geometric mean. … ►using the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute . …9: Bibliography Q
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On two problems concerning means.
J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
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