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1: 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. …

2: 27.13 Functions

… ►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. … ►This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. … ►Waring’s problem is to find, for each positive integer $k$, whether there is an integer $m$ (depending only on $k$) such that the equation … ►The exact value of $g\left(k\right)$ is now known for every $k\le 200,000$. …for all $k\ge 2$, with equality if $4\le k\le 200,000$. …

3: 8.26 Tables

… ►

Pearson (1965) tabulates the function $I(u,p)$ ($=P(p+1,u)$) 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 $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ 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 $p=-\frac{9}{2}(1)-\frac{1}{2}$, $0\le x\le 200$ to 14S.

►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ 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.

…

4: Philip J. Davis

… ►He returned to Harvard after the war and completed a Ph. … ►Davis, easing into his role as Chief, did his part to promote the field of numerical analysis at NBS. …After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions. … ►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. … ►Today the DLMF contains close to 600 2D and 3D graphs and more than 200 interactive 3D visualizations. …

5: 7.17 Inverse Error Functions

… ►

7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

ⓘ

Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

… ►For these results and 25S values of the first 200 coefficients see Strecok (1968). … ►As $x\to 0$ … ►

7.17.5 $$u=-2/\ln \left(\pi x^2\ln \left(1/x\right)\right),$$

ⓘ

Defines:

$u$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

… ►

7.17.6 $$v=\ln \left(\ln \left(1/x\right)\right)-2+\ln \pi .$$

ⓘ

Defines:

$v$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

…

6: 4.9 Continued Fractions

… ►

4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

… ►See also Cuyt et al. (2008, pp. 196–200). …

7: 34.12 Physical Applications

… ►For applications in nuclear structure, see de-Shalit and Talmi (1963); in atomic spectroscopy, see Biedenharn and van Dam (1965, pp. 134–200), Judd (1998), Sobelman (1992, Chapter 4), Shore and Menzel (1968, pp. 268–303), and Wigner (1959); in molecular spectroscopy and chemical reactions, see Burshtein and Temkin (1994, Chapter 5), and Judd (1975). …

8: DLMF Project News

error generating summary

9: 10.73 Physical Applications

… ►

10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►See Jackson (1999, Chapter 9, §9.6), Jones (1986, Chapters 7, 8), and Lord Rayleigh (1945, Vol. I, Chapter IX, §§200–211, 218, 219, 221a; Vol. II, Chapter XIII, §272a; Chapter XV, §302; Chapter XVIII; Chapter XIX, §350; Chapter XX, §357; Chapter XXI, §369). …Consequently, Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. … ►For applications of the Rayleigh function ${\sigma }_n\left(\nu \right)$ (§10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).

10: 28.35 Tables

… ►

Blanch and Rhodes (1955) includes ${\mathrm{𝐵𝑒}}_n(t)$, ${\mathrm{𝐵𝑜}}_n(t)$, $t=\frac{1}{2}\sqrt{q}$, $n=0(1)15$; 8D. The range of $t$ is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: ${\mathrm{𝐵𝑒}}_n(t)=a_n\left(q\right)+2q-(4n+2)\sqrt{q}$, ${\mathrm{𝐵𝑜}}_n(t)=b_n\left(q\right)+2q-(4n-2)\sqrt{q}$.

►

Ince (1932) includes eigenvalues $a_n$, $b_n$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also ${\mathrm{ce}}_n(x,q)$, ${\mathrm{se}}_n(x,q)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_n={\mathrm{𝑏𝑒}}_n-2q$, $b_n={\mathrm{𝑏𝑜}}_n-2q$.

… ►

National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ <span class="ltx\_text hilite">to</span> }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

… ►

Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

… ►For other tables prior to 1961 see Fletcher et al. (1962, §2.2) and Lebedev and Fedorova (1960, Chapter 11).