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  • 12: 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. …
    13: 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). …
    14: 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 ) .
    15: How to Cite
    Citations from other electronic media (the web, email, …), should, of course, use the appropriate means to give the site URL (https://dlmf.nist.gov/), or specific Permalinks. …
    16: 4.1 Special Notation
    Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsin z and Arcsin z , etc. …
    17: Bille C. Carlson
    Also, the homogeneity of the R -function has led to a new type of mean value for several variables, accompanied by various inequalities. …
    18: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 c t d = 0 ; see §1.11(iii). These roots are necessarily distinct and represent e 1 , e 2 , e 3 in some order.

    If c and d are real, and the discriminant is positive, that is c 3 27 d 2 > 0 , then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

    If c 3 27 d 2 < 0 , or c and d are not both real, then we label α , β , γ so that the triangle with vertices α , β , γ is positively oriented and [ α , γ ] is its longest side (chosen arbitrarily if there is more than one). In particular, if α , β , γ are collinear, then we label them so that β is on the line segment ( α , γ ) . In consequence, k 2 = ( β γ ) / ( α γ ) , k 2 = ( α β ) / ( α γ ) satisfy k 2 0 k 2 (with strict inequality unless α , β , γ are collinear); also | k 2 | , | k 2 | 1 .

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 k 2 ) d ( 1 k 2 k 2 ) ,

    where M denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 2 possible choices of the square root.

  • 19: 1.4 Calculus of One Variable
    Mean Value Theorem
    First Mean Value Theorem
    Second Mean Value Theorem
    20: 1.13 Differential Equations
    Then the following relation is known as Abel’s identity