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1: Sidebar 9.SB1: Supernumerary Rainbows

… ►Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …

2: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

3: Philip J. Davis

… ►At CalTech, John Todd dedicated himself to the training of new researchers in numerical analysis, and Olga Taussky, who had been a full-time NBS consultant influential in establishing the field of matrix theory, became the first woman at CalTech to attain the academic rank of full professor. … ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. … ►This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms. …

4: Leonard C. Maximon

… ►He had been an Assistant Professor in the Graduate Division of Applied Mathematics at Brown University, a Visiting Professor at the Norwegian Technical University in Trondheim, Norway, and a Physicist at the Center for Radiation Research at the National Bureau of Standards. …

5: Richard B. Paris

… ►He also had an honorary appointment at the University of St. …

6: Ingram Olkin

… ►He also was a Guggenheim, Fulbright, Humboldt, and Lady Davis Fellow, had an honorary degree from De Montfort University, and was awarded the Townsend Harris Medal by the City University of New York. …

7: About MathML

… ►The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. …

8: About the Project

… ►Whereas before the project had Editors and Associate Editors, it now has Editors, Senior Associate Editors, and Associate Editors. … ►It was fortunate that the project had already recruited Adri Olde Daalhuis from the University of Edinburgh in 2012 to serve as an additional Mathematics Editor. …

9: 18.16 Zeros

… ►

Asymptotic Behavior

… ►

18.16.12 $$(n+2)x_{n,1}\ge {\left(n-1-\sqrt{n^2+(n+2)(\alpha +1)}\right)}^2-1,$$

ⓘ

Symbols:

$n$: nonnegative integer and $x$: real variable

Referenced by:

§18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)

Permalink:

http://dlmf.nist.gov/18.16.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,1}>2n+\alpha -2-{(1+4(n-1)(n+\alpha -1))}^{\frac{1}{2}}$, which had been taken from Ismail and Li (1992).

See also:

Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

►

18.16.13 $$(n+2)x_{n,n}\le {\left(n-1+\sqrt{n^2+(n+2)(\alpha +1)}\right)}^2-1.$$

ⓘ

Symbols:

$n$: nonnegative integer and $x$: real variable

Referenced by:

§18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)

Permalink:

http://dlmf.nist.gov/18.16.E13

Encodings:

pMML, png, TeX

Modification (effective with 1.0.5):

This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound , which had been taken from Ismail and Li (1992).

See also:

Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

… ►For an error bound for the first approximation yielded by this expansion see Olver (1997b, p. 408). ►Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $L_n^{(\pm \frac{1}{2})}\left(x\right)$ lead immediately to results for the zeros of $H_n\left(x\right)$. …

10: 5.4 Special Values and Extrema

… ►As $n\to \mathrm{\infty }$, ►

5.4.20 $$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

6.3.20 (The error estimate has been improved.)

Referenced by:

§5.4(iii)

Permalink:

http://dlmf.nist.gov/5.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(iii), §5.4 and Ch.5

…