transcendental equations

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11: Bibliography P

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Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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External Links:

ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

BibTeX

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Notes:

Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

Encodings:

BibTeX

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

Encodings:

BibTeX

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12: 19.20 Special Cases

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

19.20.11 $$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$ $y\to 0+$; $p$ ($\ne 0$) real.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\sim$: Poincaré asymptotic expansion and $\ln z$: 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 $\frac{-3}{p}{R}_C(z,p)$ 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. …