Landen transformation

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1: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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§22.7(iii) Generalized Landen Transformations

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2: 22.17 Moduli Outside the Interval [0,1]

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§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 $\sqrt{k}$ and $k^{\prime }=\sqrt{1-k^2}$ 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

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§22.20(iii) Landen Transformations

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5: 20.7 Identities

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§20.7(vi) Landen Transformations

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6: 19.8 Quadratic Transformations

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§19.8(ii) Landen Transformations

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Descending Landen Transformation

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Ascending Landen Transformation

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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 (implying ), and descending Gauss transformations when (implying ). …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 $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). … ►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both $F(\varphi ,k)$ and $E(\varphi ,k)$. … ►If ${\alpha }^2=k^2$, then the method fails, but the function can be expressed by (19.6.13) in terms of $E(\varphi ,k)$, for which Neuman (1969b) uses ascending Landen transformations. … ►The function $\mathrm{el2}(x,k_c,a,b)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). …

9: Bibliography C

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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.

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External Links:

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Referenced by:

§19.22(iii).

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