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11: 1.5 Calculus of Two or More Variables

… ►A function is continuous on a point set

D

if it is continuous at all points of

D

. … ►If

f(x,y)

is continuous, and

D

is the set … ►Similarly, if

D

is the set … ►If

D

can be represented in both forms (1.5.30) and (1.5.33), and

f(x,y)

is continuous on

D

, then … ►Infinite double integrals occur when

f(x,y)

becomes infinite at points in

D

or when

D

is unbounded. …

12: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all

n\leq 10^{4}

: (a) the canonical factorization of

n

into powers of primes; (b) the Euler totient

\phi\left(n\right)

; (c) the divisor function

d\left(n\right)

; (d) the sum

\sigma(n)

of these divisors. …Table III lists all solutions

n\leq 10^{4}

of the equation

d\left(n\right)=m

, and Table IV lists all solutions

n

of the equation

\sigma(n)=m

for all

m\leq 10^{4}

. …6 lists

\phi\left(n\right),d\left(n\right)

, and

\sigma(n)

for

n\leq 1000

; Table 24. …

13: 12.1 Special Notation

… ►An older notation, due to Whittaker (1902), for

U\left(a,z\right)

D_{\nu}\left(z\right)

. The notations are related by

U\left(a,z\right)=D_{-a-\frac{1}{2}}\left(z\right)

. Whittaker’s notation

D_{\nu}\left(z\right)

is useful when

\nu

is a nonnegative integer (Hermite polynomial case).

14: 21.6 Products

… ►

21.6.2

\mathcal{D}=|\mathbf{T}^{\mathrm{T}}{\mathbb{Z}}^{h}/(\mathbf{T}^{\mathrm{T}}{% \mathbb{Z}}^{h}\cap{\mathbb{Z}}^{h})|,

ⓘ

Defines:: $\mathcal{D}$ : number of elements (locally)
Symbols:: $\mathbb{Z}$ : set of all integers, $\cap$ : intersection, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix and $h$ : positive integer
Permalink:: http://dlmf.nist.gov/21.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6 and Ch.21

►that is,

\mathcal{D}

is the number of elements in the set containing all

h

-dimensional vectors obtained by multiplying

\mathbf{T}^{\mathrm{T}}

on the right by a vector with integer elements. … ►

21.6.3

\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}\in% \mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i\operatorname{tr}\left[% \frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{% \mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}^{h}\theta\left(\mathbf% {z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{% {\Omega}}\right),

… ►

21.6.4

\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\sum_{k=1}^{h}T_{jk}\mathbf{c}_{k% }}{\sum_{k=1}^{h}T_{jk}\mathbf{d}_{k}}\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}% \middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}% \in\mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{-2\pi i\sum_{j=1}^{h}\mathbf{% b}_{j}\cdot\mathbf{c}_{j}}\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\mathbf% {a}_{j}+\mathbf{c}_{j}}{\mathbf{b}_{j}+\mathbf{d}_{j}}\left(\mathbf{z}_{j}% \middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $g$ : positive integer, $h$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{jk}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6 and Ch.21

… ►

21.6.6

\theta\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}% -\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x% }-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in% \frac{1}{2}{\mathbb{Z}}^{g}/{\mathbb{Z}}^{g}}\,\sum_{\boldsymbol{{\beta}}\in% \frac{1}{2}{\mathbb{Z}}^{g}/{\mathbb{Z}}^{g}}e^{2\pi i\left(2\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{% \alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}\*\theta% \left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta% }}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{y}+\boldsymbol{{% \Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}% }\right)\theta\left(\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{v}% +\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|% \boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

…

15: 24.10 Arithmetic Properties

… ►

24.10.1

B_{2n}+\sum_{(p-1)\mathbin{|}2n}\frac{1}{p}=\hbox{integer},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $p$ : prime
Permalink:: http://dlmf.nist.gov/24.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(i), §24.10 and Ch.24

… ►Let

B_{2n}=\ifrac{N_{2n}}{D_{2n}}

, with

N_{2n}

and

D_{2n}

relatively prime and

D_{2n}>0

. … ►

24.10.7

(b^{2n}-1)N_{2n}\equiv{2nb^{2n-1}D_{2n}\sum_{k=1}^{M-1}k^{2n-1}\left\lfloor% \frac{kb}{M}\right\rfloor\pmod{M}},

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\equiv$ : modular equivalence, $k$ : integer, $n$ : integer, $M$ : integer and $b$ : integer
Permalink:: http://dlmf.nist.gov/24.10.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iii), §24.10 and Ch.24

…

16: 19.29 Reduction of General Elliptic Integrals

… ►where the arguments of the

R_{D}

function are, in order,

(a-b)(u-c)

(b-c)(a-u)

(a-b)(b-c)

. … ►The only cases that are integrals of the third kind are those in which at least one

m_{j}

with

j>h

is a negative integer and those in which

h=4

and

\sum_{j=1}^{n}m_{j}

is a positive integer. … ►where … ►

V=\sqrt{D_{12}^{2}-D_{11}D_{22}}.

… ►

D_{11}=-b_{1}^{2}.

…

17: 19.20 Special Cases

… ►

R_{J}\left(x,y,z,z\right)=R_{D}\left(x,y,z\right),

… ►

§19.20(iv) $R_{D}\left(x,y,z\right)$

… ►

R_{D}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-3/2}R_{D}\left(x,y,z% \right),

… ►

R_{D}\left(0,0,z\right)=\infty.

… ►Define

c=\sum_{j=1}^{n}b_{j}

. …

18: 24.19 Methods of Computation

… ►

D_{2n}=\prod_{p-1\mathbin{|}2n}p,

►

B_{2n}=\dfrac{N_{2n}}{D_{2n}}.

… ►

•

Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

►

•

Buhler et al. (1992) uses the expansion

24.19.3

\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii)
Permalink:: http://dlmf.nist.gov/24.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

…

19: 27.2 Functions

… ►It can be expressed as a sum over all primes

p\leq x

: … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Assume that

\mathcal{D}(T)

is dense in

V

, i. … ►

u_{\lambda}\in\mathcal{D}(T)

, corresponding to distinct eigenvalues, are orthogonal: i. … ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that

\mathcal{D}(T)=\mathcal{D}({T}^{*})

, as

f(x)

and

g(x)

satisfy the same boundary conditions and thus define the same domains. … ►,

\mathcal{D}(T)\subset\mathcal{D}({T}^{*})

and

{T}^{*}v=Tv

for

v\in\mathcal{D}(T)

. … ► We have a direct sum of linear spaces:

\mathcal{D}({T}^{*})=\mathcal{D}(T^{**})+N_{\mathrm{i}}+N_{-\mathrm{i}}

. …