Landen%20transformations

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1: 19.36 Methods of Computation

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§19.36(ii) Quadratic 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 $F(\varphi ,k)$ 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)). …

2: 20.7 Identities

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§20.7(iv) Reduction Formulas for Products

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§20.7(vi) Landen Transformations

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(viii) Transformations of Lattice Parameter

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§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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3: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

Encodings:

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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