# Landen transformations

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## 10 matching pages

##### 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 $\sqrt{k}$ 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). … …
##### 5: 20.7 Identities
###### Ascending LandenTransformation
If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when $x,y (implying $a), and descending Gauss transformations when $z (implying $z_{+}). …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. …
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 $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$. … If $\alpha^{2}=k^{2}$, then the method fails, but the function can be expressed by (19.6.13) in terms of $E\left(\phi,k\right)$, for which Neuman (1969b) uses ascending Landen transformations. … The function $\mathrm{el2}\left(x,k_{c},a,b\right)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). …
• G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.