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1: Browsers

… ►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.
…
$a\circ b$	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 $-1$ .
…

…

3: Guide to Searching the DLMF

… ►Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. …

4: 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

…

5: 19.22 Quadratic Transformations

… ►Let

\Re x>0

\Re y>0

a=(x+y)/2

, and

p\neq 0

. … ►If

p=y

, then (19.22.7) reduces to (19.22.3), but if

p=x

p=y

, then both sides of (19.22.4) are 0 by (19.20.9). … ►(If

p_{0}=a_{0}

, then

p_{n}=a_{n}

and (19.22.13) reduces to (19.22.11).) … ►If

p=x

p=y

, then (19.22.20) reduces to

0=0

by (19.20.13), and if

z=x

z=y

then (19.22.19) reduces to

0=0

by (19.20.20) and (19.22.22). …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

. …

6: 1.11 Zeros of Polynomials

… ►Add

-\tfrac{1}{4}a

to the roots of

g(w)=0

to get those of

f(z)=0

. …

7: 7.1 Special Notation

… ►Alternative notations are

Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)

P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)

\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z

\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)

C_{1}(z)=C\left(\sqrt{2/\pi}z\right)

S_{1}(z)=S\left(\sqrt{2/\pi}z\right)

C_{2}(z)=C\left(\sqrt{2z/\pi}\right)

S_{2}(z)=S\left(\sqrt{2z/\pi}\right)

. …

8: 27.17 Other Applications

… ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …

9: 19.25 Relations to Other Functions

… ►All terms on the right-hand sides are nonnegative when

k^{2}\leq 0

0\leq k^{2}\leq 1

, or

1\leq k^{2}\leq c

, respectively. … ►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). … ►Let

r=1/x^{2}

. … ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which

\wp\left(z\right)-e_{j}<0

, for some

j

. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which

\sigma_{j}^{2}(z)<0

, for some

j

. …

10: 5.1 Special Notation

… ►Alternative notations for this function are:

\Pi(z-1)

(Gauss) and

(z-1)!

. Alternative notations for the psi function are:

\Psi(z-1)

(Gauss) Jahnke and Emde (1945);

\Psi(z)

Davis (1933);

\mathsf{F}(z-1)

Pairman (1919). …