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1: 19.8 Quadratic Transformations
§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
As n , 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. … Again, p n and ε 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
Abel Summability
Abel Means
A ( r , θ ) is a harmonic function in polar coordinates ((1.9.27)), and … If f ( θ ) is periodic and integrable on [ 0 , 2 π ] , then as n the Abel means A ( r , θ ) and the (C,1) means σ n ( θ ) converge to …
Abel Summability
3: 1.2 Elementary Algebra
§1.2(iv) Means
The arithmetic mean of n numbers a 1 , a 2 , , a n is … The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , 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 , , a n , and n positive numbers p 1 , p 2 , , p n with …
M ( 1 ) = A ,
4: 22.18 Mathematical Applications
§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). …
5: 2.10 Sums and Sequences
Another version is the Abel–Plana formula: …
  • (c)

    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: 1.7 Inequalities
    §1.7(iii) Means
    1.7.7 H G A ,
    1.7.8 min ( a 1 , a 2 , , a n ) M ( r ) max ( a 1 , a 2 , , a n ) ,
    1.7.9 M ( r ) M ( s ) , r < s ,
    7: 15.17 Mathematical Applications
    Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
    8: Bibliography Q
  • S.-L. Qiu and J.-M. Shen (1997) On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
  • 9: Bibliography B
  • H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
  • A. I. Bobenko (1991) Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).
  • 10: Need Help?
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