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1: 25.10 Zeros

… в–єCalculations relating to the zeros on the critical line make use of the real-valued function …where … в–єSign changes of

Z(t)

are determined by multiplying (25.9.3) by

\exp\left(i\vartheta(t)\right)

to obtain the Riemann–Siegel formula: …where

R(t)=O\left(t^{-1/4}\right)

t\to\infty

. … в–єMore than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

2: Sidebar 7.SB1: Diffraction from a Straightedge

… в–єThe intensity distribution follows

|\mathcal{F}\left(x\right)|^{2}

, where

\mathcal{F}

is the Fresnel integral (See 7.3.4). …The faint circular patterns are additional diffraction effects due to imperfections in the edge.

… в–єOne source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). …

4: Stephen M. Watt

… в–єCheriton School of Computer Science at the University of Waterloo, where from 2015 to 2020 he also served as Dean of the Faculty of Mathematics. … в–єPrior to joining the University of Waterloo, Watt was Distinguished University Professor of the University of Western Ontario and Professor at the University of Nice-Sophia Antipolis. …

5: Barry I. Schneider

… в–єBefore coming to NIST in 2014, he was a postdoctoral research associate at the University of Southern California (1969-1970), and a staff member of the General Telephone and Electronics Laboratory (1970-1972). He joined the Theoretical Division of the Los Alamos National Laboratory (1972-1991) and then the National Science Foundation (1991-2013) where he was a Program Director in the Physics Division and then in the Office of Cyberinfrastructure. In early 2014, he came to NIST as General Editor of the DLMF project. … в–єHe was a visiting scientist at NIST from 1995 to 2013 and spent a sabbatical year at NIST in 2000-2001. …Recently he has served as Co-Chair of the US government Fast Track Action Committee to update the US strategic computing plan.

6: Joyce E. Conlon

… в–єIn 1999 she joined the NIST Mathematical and Computational Sciences Division, where she contributed to the DLMF project, especially in the construction of the bibliography. …

7: Tom M. Apostol

… в–єHe was a visiting professor at the University of Patras in Greece in 1978, and was elected a Corresponding Member of the Academy of Athens in 2001 (where he delivered his inaugural lecture in Greek). … в–єHe was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … в–єIn 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … Ford Award, given to recognize authors of articles of expository excellence. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

8: 5.10 Continued Fractions

… в–є

5.10.1

\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,

в“

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $a_{k}$ : coefficient
Referenced by:: Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
Permalink:: http://dlmf.nist.gov/5.10.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.10 and Ch.5

в–єwhere …For exact values of

a_{7}

a_{11}

and 40S values of

a_{0}

a_{40}

, see Char (1980). …

9: 27.15 Chinese Remainder Theorem

… в–єThe Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. в–єThis theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers

a_{1}\pmod{m_{1}}

a_{2}\pmod{m_{2}}

a_{3}\pmod{m_{3}}

, and

a_{4}\pmod{m_{4}}

, where each

a_{j}

has no more than five digits. These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

10: 17.17 Physical Applications

… в–єThey were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. … в–єIt involves

q

-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. See Micu and Papp (2005), where many earlier references are cited.