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21: 1.13 Differential Equations

… ►Here dots denote differentiations with respect to

\zeta

, and

\left\{z,\zeta\right\}

is the Schwarzian derivative: … ►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). … ►The functions

u(x)

which correspond to these being eigenfunctions. … ►

Transformation to Liouville normal Form

►Equation (1.13.26) with

x\in[a,b]

may be transformed to the Liouville normal form …

22: 19.36 Methods of Computation

… ►In the Appendix of the last reference it is shown how to compute

R_{J}

without computing

R_{C}

more than once. … ►As

n\to\infty

c_{n}

a_{n}

, and

t_{n}

converge quadratically to limits

0

M

, and

T

, respectively; hence … ►To (19.36.6) add … ►(19.22.20) reduces to

0=0

p=x

p=y

, and (19.22.19) reduces to

0=0

z=x

z=y

. … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …

23: 19.29 Reduction of General Elliptic Integrals

… ►

§19.29(i) Reduction Theorems

►These theorems reduce integrals over a real interval

(y,x)

of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over

(0,\infty)

containing the square root of a cubic polynomial (compare §19.16(i)). … ►

§19.29(ii) Reduction to Basic Integrals

… ►Partial fractions provide a reduction to integrals in which

\mathbf{m}

has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. …which shows how to express the basic integral

I(-\mathbf{e}_{j})

in terms of symmetric integrals by using (19.29.4) and either (19.29.7) or (19.29.8). …

24: Philip J. Davis

… ►He returned to Harvard after the war and completed a Ph. … ►In 1957, Davis took over as Chief, Numerical Analysis Section when John Todd and his wife Olga Taussky-Todd, feeling a strong pull toward teaching and research, left to pursue full-time positions at the California Institute of Technology. … ►After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions. … ►NBS mathematician Irene Stegun took over management of the A&S project which was already well on its way, and led the work to publication in 1964. … ►Today the DLMF contains close to 600 2D and 3D graphs and more than 200 interactive 3D visualizations. …

25: 8.26 Tables

… ►

•

Pearson (1965) tabulates the function $I(u,p)$ ( $=P\left(p+1,u\right)$ ) for $p=-1(.05)0(.1)5(.2)50$ , $u=0(.1)u_{p}$ to 7D, where $I(u,u_{p})$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$ , $u=0(.1)6$ to 5D.

… ►

•

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.

►

•

Chiccoli et al. (1988) presents a short table of $E_{p}\left(x\right)$ for $p=-\tfrac{9}{2}(1)-\tfrac{1}{2}$ , $0\leq x\leq 200$ to 14S.

►

•

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.

►

•

Stankiewicz (1968) tabulates $E_{n}\left(x\right)$ for $n=1(1)10$ , $x=0.01(.01)5$ to 7D.

…

26: About Color Map

… ►We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. … ►To provide an easily interpreted encoding of surface heights, a rainbow-like mapping of height to color is used. The following figure illustrates the piece-wise linear mapping of the height to each of the color components red, green and blue, written as

\left\langle R,\;G,\;B\right\rangle

. … ►In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of

\pi/2

correponding to the real and imaginary axes, at more immediately recognizable colors. … ►We therefore use a piecewise linear mapping as illustrated below, that takes phase

0

to red,

\pi/2

to yellow,

\pi

to cyan and

3\pi/2

to blue. …

27: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►In April 2011, NIST co-organized a conference on “Special Functions in the 21st Century: Theory & Application” which was dedicated to Olver. … ►Olver continued to maintain a connection to NIST after moving to the university. …Fitting capstones to Olver’s long career, these works are on track to extend the legacy of the classic Abramowitz and Stegun handbook well into the 21st century. According to DLMF Project Lead Daniel Lozier, “Olver’s encyclopedic knowledge of the field, his clear vision for mathematical exposition, his keen sense of the needs of practitioners, and his unfailing attention to detail were key to the success of that project. …

28: Possible Errors in DLMF

… ►One 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). …

29: 24.20 Tables

… ►In Wagstaff (2002) these results are extended to

n=60(2)152

and

n=40(2)88

, respectively, with further complete and partial factorizations listed up to

n=300

and

n=200

, respectively. …

30: Bonita V. Saunders

… ►In 1985 she was the first African American and first woman to obtain a Ph. … ►In 2001 she was selected by the National Association of Mathematicians (NAM) to present the Claytor Lecture at the Joint Mathematics Meetings in New Orleans in memory of W. …Schieffelin Claytor, one of the first African Americans to publish mathematical research beyond the dissertation. … ►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …