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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
Cesàro Summability
General Cesàro Summability
Cesàro (or (C,1)) Means
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 …
Cesàro 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: 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 ,
5: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
6: 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).
  • 7: Need Help?
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  • 8: 19.22 Quadratic Transformations
    §19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
    The AGM, M ( a 0 , g 0 ) , of two positive numbers a 0 and g 0 is defined in §19.8(i). …
    19.22.9 4 π R G ( 0 , a 0 2 , g 0 2 ) = 1 M ( a 0 , g 0 ) ( a 0 2 - n = 0 2 n - 1 c n 2 ) = 1 M ( a 0 , g 0 ) ( a 1 2 - n = 2 2 n - 1 c n 2 ) ,
    As n , 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. …
    9: 8.25 Methods of Computation
    See Allasia and Besenghi (1987b) for the numerical computation of Γ ( a , z ) from (8.6.4) by means of the trapezoidal rule. … The computation of γ ( a , z ) and Γ ( a , z ) by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). …
    10: 7.20 Mathematical Applications
    The normal distribution function with mean m and standard deviation σ is given by
    7.20.1 1 σ 2 π - x e - ( t - m ) 2 / ( 2 σ 2 ) d t = 1 2 erfc ( m - x σ 2 ) = Q ( m - x σ ) = P ( x - m σ ) .