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21—30 of 827 matching pages

21: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that …Here

\xi(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. … ►

21.5.5

\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{-1}]^{\mathrm{T}}\end{bmatrix}\Rightarrow% \theta\left(\mathbf{A}\mathbf{z}\middle|\mathbf{A}\boldsymbol{{\Omega}}\mathbf% {A}^{\mathrm{T}}\right)=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

►(

\mathbf{A}

invertible with integer elements.) …For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. …

22: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

§8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

… ►For an exponentially-improved asymptotic expansion of

E_{p}\left(z\right)

see §2.11(iii). … ►For

x\geq 0

and

p>1

let

x=\lambda p

and define

A_{0}(\lambda)=1

, …so that

A_{k}(\lambda)

is a polynomial in

\lambda

of degree

k-1

when

k\geq 1

. … ►

A_{3}(\lambda)=1-8\lambda+6\lambda^{2}.

…

23: 1.12 Continued Fractions

… ►

C_{n}

is called the

n

th approximant or convergent to $C$ .

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►Define … ►Conversely,

C

is called an extension of

C^{\prime}

. … ►Then the convergents

C_{n}

satisfy …

24: 23.15 Definitions

… ►Also

\mathcal{A}

denotes a bilinear transformation on

\tau

, given by … ►A modular function

f(\tau)

is a function of

\tau

that is meromorphic in the half-plane

\Im\tau>0

, and has the property that for all

\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})

, or for all

\mathcal{A}

belonging to a subgroup of SL

(2,\mathbb{Z})

, ►

23.15.5

f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),

\Im\tau>0

ⓘ

Symbols:: $\Im$ : imaginary part, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $c$ : integer, $d$ : integer, $f(\tau)$ : modular function, $c_{\mathcal{A}}$ : constant and $\ell$ : level
Permalink:: http://dlmf.nist.gov/23.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(i), §23.15 and Ch.23

►where

c_{\mathcal{A}}

is a constant depending only on

\mathcal{A}

, and

\ell

(the level) is an integer or half an odd integer. … ►

23.15.7

J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},

ⓘ

Defines:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

…

25: 26.10 Integer Partitions: Other Restrictions

… ►The set

\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}

is denoted by

A_{j,k}

. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). … ►where

I_{1}\left(x\right)

is the modified Bessel function (§10.25(ii)), and …The quantity

A_{k}(n)

is real-valued. …

26: 3.7 Ordinary Differential Equations

… ►where

f

g

, and

h

are analytic functions in a domain

D\subset\mathbb{C}

. … ►where

\mathbf{A}(\tau,z)

is the matrix … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. …

27: 1.1 Special Notation

… ► ►►►►►►

$x, y$	real variables.
…
$\mathbf{A}$	or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$ .
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
…
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
$\operatorname{tr}(\mathbf{A})$	trace of the square matrix $\mathbf{A}$
…

►In the physics, applied maths, and engineering literature a common alternative to

\overline{a}

a^{*}

a

being a complex number or a matrix; the Hermitian conjugate of

\mathbf{A}

is usually being denoted

\mathbf{A}^{{\dagger}}

28: 3.3 Interpolation

… ►If

f

is analytic in a simply-connected domain

D

(§1.13(i)), then for

z\in D

, …where

C

is a simple closed contour in

D

described in the positive rotational sense and enclosing the points

z,z_{1},z_{2},\dots,z_{n}

. … ►and

A_{k}^{n}

are the Lagrangian interpolation coefficients defined by … ►where

\omega_{n+1}(\zeta)

is given by (3.3.3), and

C

is a simple closed contour in

{D}

described in the positive rotational sense and enclosing

z_{0},z_{1},\dots,z_{n}

. … ►Then by using

x_{3}

in Newton’s interpolation formula, evaluating

\left[x_{0},x_{1},x_{2},x_{3}\right]f=-0.26608\;28233

and recomputing

f^{\prime}(x)

, another application of Newton’s rule with starting value

x_{3}

gives the approximation

x=2.33810\;7373

, with 8 correct digits. …

29: 28.31 Equations of Whittaker–Hill and Ince

… ►and constant values of

A, B, k

, and

c

, is called the Equation of Whittaker–Hill. … ►When

k^{2}<0

, we substitute … ►They are denoted by … ►They are real and distinct, and can be ordered so that

C_{p}^{m}(z,\xi)

and

S_{p}^{m}(z,\xi)

have precisely

m

zeros, all simple, in

0\leq z<\pi

. …ambiguities in sign being resolved by requiring

C_{p}^{m}(x,\xi)

and

{S_{p}^{m}}^{\prime}(x,\xi)

to be continuous functions of

x

and positive when

x=0

. …

30: 19.25 Relations to Other Functions

… ►Let

\mathbb{L}

be a lattice for the Weierstrass elliptic function

\wp\left(z\right)

. …for some

2\omega\in\mathbb{L}

, provided that

z

satisfies … ►for some

2\omega_{j}\in\mathbb{L}

and

\wp\left(\omega_{j}\right)=e_{j}

. … ►for some

2\omega\in\mathbb{L}

, where …in which

2\omega_{1}

and

2\omega_{3}

are generators for the lattice

\mathbb{L}

\omega_{2}=-\omega_{1}-\omega_{3}

, and

\eta_{j}=\zeta\left(\omega_{j}\right)

(see (23.2.12)). …

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21—30 of 827 matching pages

21: 21.5 Modular Transformations

22: 8.20 Asymptotic Expansions of E p ⁡ ( z )

§8.20 Asymptotic Expansions of E p ⁡ ( z )

23: 1.12 Continued Fractions

24: 23.15 Definitions

25: 26.10 Integer Partitions: Other Restrictions

26: 3.7 Ordinary Differential Equations

27: 1.1 Special Notation

28: 3.3 Interpolation

29: 28.31 Equations of Whittaker–Hill and Ince

30: 19.25 Relations to Other Functions

22: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

§8.20 Asymptotic Expansions of $E_{p}\left(z\right)$