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1: 22.7 Landen Transformations
§22.7(ii) Ascending Landen Transformation
2: 6.13 Zeros
Ci ( x ) and si ( x ) each have an infinite number of positive real zeros, which are denoted by c k , s k , respectively, arranged in ascending order of absolute value for k = 0 , 1 , 2 , . …
3: 22.17 Moduli Outside the Interval [0,1]
§22.17(ii) Complex Moduli
4: 19.8 Quadratic Transformations
Ascending Landen Transformation
We consider only the descending Gauss transformation because its (ascending) inverse moves F ( ϕ , k ) closer to the singularity at k = sin ϕ = 1 . …
5: 19.22 Quadratic Transformations
If x , y , z are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when x , y < z (implying a < z - < z + ), and descending Gauss transformations when z < x , y (implying z + < z - < a ). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …
6: 3.10 Continued Fractions
if the expansion of its n th convergent C n in ascending powers of z agrees with (3.10.7) up to and including the term in z n - 1 , n = 1 , 2 , 3 , . … We say that it is associated with the formal power series f ( z ) in (3.10.7) if the expansion of its n th convergent C n in ascending powers of z , agrees with (3.10.7) up to and including the term in z 2 n - 1 , n = 1 , 2 , 3 , . …
7: 4.45 Methods of Computation
The function ln x can always be computed from its ascending power series after preliminary scaling. … The function arctan x can always be computed from its ascending power series after preliminary transformations to reduce the size of x . …
8: 22.20 Methods of Computation
§22.20(iii) Landen Transformations
9: 30.16 Methods of Computation
and real eigenvalues α 1 , d , α 2 , d , , α d , d , arranged in ascending order of magnitude. …
10: 9.9 Zeros
They are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . They lie in the sectors 1 3 π < ph z < 1 2 π and - 1 2 π < ph z < - 1 3 π , and are denoted by β k , β k , respectively, in the former sector, and by β k ¯ , β k ¯ , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for k = 1 , 2 , . See §9.3(ii) for visualizations. …