transcendental equations

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11—12 of 12 matching pages

11: Bibliography C

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R. Campbell (1955) Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.
Encodings:: BibTeX

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CEPHES (free C library)

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Notes:: This library treats about 180 different mathematical functions, including a substantial collection of probability integrals, Bessel functions, and higher transcendental functions. Implementations in single, double, and quad precisions are provided. See Moshier (1989).
External Links:: GAMS (package CEPHES)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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T. W. Chaundy (1969) Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:: ISBN 0-19-853139-7, MathReview, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

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P. A. Clarkson (2003a) The third Painlevé equation and associated special polynomials. J. Phys. A 36 (36), pp. 9507–9532.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iii), §32.9(i).
Encodings:: BibTeX

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P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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12: 19.20 Special Cases

… ► Schneider that this is a transcendental number. … ►

19.20.11

R_{J}\left(0,y,z,p\right)=\frac{3}{2p\sqrt{z}}\ln\left(\frac{16z}{y}\right)-% \frac{3}{p}R_{C}\left(z,p\right)+O\left(y\ln y\right),

y\to 0+

;

p

(

\neq 0

) real.

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\sim$ : Poincaré asymptotic expansion and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), Erratum (V1.0.26) for Equation (19.20.11)
Permalink:: http://dlmf.nist.gov/19.20.E11
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.26):: The constant term $\frac{-3}{p}R_{C}\left(z,p\right)$ and the order term $O\left(y\ln y\right)$ were added, and hence $\sim$ was replaced by $=$ .
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

… ► Schneider that this is a transcendental number. …