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1: Browsers

… ►Although we are not yet encoding our graphics in SVG, will be looking to make that conversion in the future. … ►Also see About MathML for help for solving problems with MathML in your browser. …Although we have attempted to follow standards and maintain backwards compatibility with older browsers, you will generally get the best results by upgrading to the latest version of your preferred browser.

2: 21.1 Special Notation

… ► ►►►►►►

$g, h$	positive integers.
…
$\boldsymbol{{\alpha}},\boldsymbol{{\beta}}$	$g$ -dimensional vectors, with all elements in $[0,1)$ , unless stated otherwise.
…
$S^{g}$	set of $g$ -dimensional vectors with elements in $S$ .
…
$S_{1}/S_{2}$	set of all elements of $S_{1}$ , modulo elements of $S_{2}$ . Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$ . (For an example see §20.12(ii).)
$a\circ b$	intersection index of $a$ and $b$ , two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$ .
…

… ►The main functions treated in this chapter are the Riemann theta functions

\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

, and the Riemann theta functions with characteristics

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

. …

3: Guide to Searching the DLMF

… ►In DLMF, you can search not only for text but also for mathematical expressions. Every page contains a search box in the navigation bar. … ►

Fine Points in Math Search

… ►Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. … ►In the following situations, DLMF search is case-sensitive: …

4: 19.22 Quadratic Transformations

… ►

19.22.3

2y^{2}R_{D}\left(0,x^{2},y^{2}\right)=\tfrac{1}{4}(y^{2}-x^{2})R_{D}\left(0,xy% ,a^{2}\right)+3R_{F}\left(0,xy,a^{2}\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.22(i), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

… ►The AGM,

M\left(a_{0},g_{0}\right)

, of two positive numbers

a_{0}

and

g_{0}

is defined in §19.8(i). Again, we assume that

a_{0}\geq g_{0}

(except in (19.22.10)), and define

c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}

. … ►Ascent and descent correspond respectively to increase and decrease of

k

in Legendre’s notation. … ►However, if

x

and

y

are complex conjugates and

z

and

p

are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and

p_{\pm}^{2}-p_{\mp}^{2}=\pm|p^{2}-x^{2}|\neq 0

. …

5: 1.11 Zeros of Polynomials

… ►With

b_{k}

as in (1.11.1)–(1.11.3) let

c_{n}=a_{n}

and … ►A similar relation holds for the changes in sign of the coefficients of

f(-z)

, and hence for the number of negative zeros of

f(z)

. … ►Add

-\tfrac{1}{4}a

to the roots of

g(w)=0

to get those of

f(z)=0

. …

6: 4.10 Integrals

… ►

4.10.5

\int_{0}^{1}\frac{\ln t}{1-t}\,\mathrm{d}t=-\frac{\pi^{2}}{6},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 4.1.55
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.10(i), §4.10 and Ch.4

►

4.10.6

\int_{0}^{1}\frac{\ln t}{1+t}\,\mathrm{d}t=-\frac{\pi^{2}}{12},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 4.1.56
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.10(i), §4.10 and Ch.4

… ►The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). … ►For

a,b\neq 0

, … ►

4.10.10

\int\frac{e^{az}-1}{e^{az}+1}\,\mathrm{d}z=\frac{2}{a}\ln\left(e^{az/2}+e^{-az% /2}\right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $a\neq 0$ : complex
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.10(ii), §4.10 and Ch.4

…

7: 19.25 Relations to Other Functions

… ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … ►for some

2\omega\in\mathbb{L}

, provided that

z

satisfies …in which the sign and the

\omega

are the same as in (19.25.35). ►In (19.25.38) and (19.25.39)

j

k

\ell

is any permutation of the numbers

1,2,3

. … ►for some

2\omega\in\mathbb{L}

, where …

8: Ira Gessel

… ► 1951 in Philadelphia, Pennsylvania) is Professor Emeritus in the Department of Mathematics at Brandeis University in Waltham, Massachusetts. …from the Massachusetts Institute of Technology in 1977. Gessel has published numerous papers in enumerative combinatorics. … ►In November 2015, Gessel was named Associate Editor of the following DLMF Chapter …

9: Ian J. Thompson

Profile
Ian J. Thompson

… ►Ian J. Thompson (b. 1953 in New Zealand) has been since 2006 a Theoretical Nuclear Physicist in the Nuclear Theory and Modeling Group of the Lawrence Livermore National Laboratory, Livermore, California. …at Auckland, New Zealand in 1979. ►Thompson has published papers on special functions, and numerous papers in theoretical nuclear physics, especially in scattering theory. …

10: Brian R. Judd

… ► 1931 in Chelmsford, U. … Dieke Professor Emeritus in the Department of Physics and Astronomy at The Johns Hopkins University, Baltimore. ►Judd’s books include Operator Techniques in Atomic Spectroscopy, published by McGraw-Hill in 1963 and reprinted by Princeton University Press in 1998, Second Quantization and Atomic Spectroscopy, published by Johns Hopkins in 1967, Topics in Atomic and Nuclear Theory (with J. … Elliott), published by Caxton Press in 1971, and Angular Momentum Theory for Diatomic Molecules, published by Academic Press in 1975. … ►Judd served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions.