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Landen transformations

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1: 22.7 Landen Transformations
§22.7 Landen Transformations
§22.7(i) Descending Landen Transformation
§22.7(ii) Ascending Landen Transformation
§22.7(iii) Generalized Landen Transformations
2: 22.17 Moduli Outside the Interval [0,1]
§22.17(ii) Complex Moduli
In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of k , irrespective of which values of k and k = 1 - k 2 are chosen—as long as they are used consistently. …
3: 19.15 Advantages of Symmetry
Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … …
4: 22.20 Methods of Computation
§22.20(iii) Landen Transformations
5: 20.7 Identities
§20.7(vi) Landen Transformations
6: 19.8 Quadratic Transformations
§19.8(ii) Landen Transformations
Descending Landen Transformation
Ascending Landen Transformation
7: 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. …
8: 19.36 Methods of Computation
The step from n to n + 1 is an ascending Landen transformation if θ = 1 (leading ultimately to a hyperbolic case of R C ) or a descending Gauss transformation if θ = - 1 (leading to a circular case of R C ). … Thompson (1997, pp. 499, 504) uses descending Landen transformations for both F ( ϕ , k ) and E ( ϕ , k ) . … If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … The function el2 ( x , k c , a , b ) is computed by descending Landen transformations if x is real, or by descending Gauss transformations if x is complex (Bulirsch (1965b)). …
9: Bibliography C
  • B. C. Carlson (1990) Landen Transformations of Integrals. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Appl. Math., Vol. 124, pp. 75–94.
  • 10: Bibliography
  • M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.
  • G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π , and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.
  • W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.
  • G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.