number-theoretic%20significance

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21—30 of 120 matching pages

21: Software Index

	Open Source	With Book	Commercial
…
20 Theta Functions
…

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

22: 8 Incomplete Gamma and Related
Functions

…

23: 28 Mathieu Functions and Hill’s Equation

…

24: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

25: 23 Weierstrass Elliptic and Modular
Functions

…

26: 2.11 Remainder Terms; Stokes Phenomenon

… ►If the results agree within

S

significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least

S

correct significant figures for larger values of

x

. … ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of

\operatorname{ph}z=\pi

, is increasingly significant as

\operatorname{ph}z

passes from

\pi

\frac{3}{2}\pi

, and becomes the dominant contribution after

\operatorname{ph}z

passes

\frac{3}{2}\pi

. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

27: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

28: 36 Integrals with Coalescing Saddles

…

29: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

30: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …