value at T=0

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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

Value at $\mathbf{T}=\boldsymbol{{0}}$

…

3: 3.11 Approximation Techniques

… ►with initial values

T_{0}\left(x\right)=1

T_{1}\left(x\right)=x

. … ►For the expansion (3.11.11), numerical values of the Chebyshev polynomials

T_{n}\left(x\right)

can be generated by application of the recurrence relation (3.11.7). … ►There exists a unique solution of this minimax problem and there are at least

k+\ell+2

values

x_{j}

a\leq x_{0}<x_{1}<\cdots<x_{k+\ell+1}\leq b

, such that

m_{j}=m

, where … ►requires

f={\mathscr{L}}^{-1}F

to be obtained from numerical values of

F

. … ►Given

n+1

distinct points

x_{k}

in the real interval

[a,b]

, with (

a=

)

x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}

(

=b

), on each subinterval

[x_{k},x_{k+1}]

k=0,1,\ldots,n-1

, a low-degree polynomial is defined with coefficients determined by, for example, values

f_{k}

and

f_{k}^{\prime}

of a function

f

and its derivative at the nodes

x_{k}

and

x_{k+1}

. …

4: 21.5 Modular Transformations

… ►

21.5.2

\boldsymbol{{\Gamma}}\mathbf{J}_{2g}\boldsymbol{{\Gamma}}^{\mathrm{T}}=\mathbf% {J}_{2g}.

ⓘ

Symbols:: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $g$ : positive integer and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

… ►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

… ►where the square root assumes its principal value. … ►For explicit results in the case

g=1

, see §20.7(viii).

5: 3.7 Ordinary Differential Equations

… ►The path is partitioned at

P+1

points labeled successively

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

, with

z_{0}=a

z_{P}=b

. … ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►If the solution

w(z)

that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along

\mathscr{P}

from

a

b

, then

w(z)

and

w^{\prime}(z)

may be computed in a stable manner for

z=z_{0},z_{1},\dots,z_{P}

by successive application of (3.7.5) for

j=0,1,\dots,P-1

, beginning with initial values

w(a)

and

w^{\prime}(a)

. … ►Similarly, if

w(z)

is decaying at least as fast as all other solutions along

\mathscr{P}

, then we may reverse the labeling of the

z_{j}

along

\mathscr{P}

and begin with initial values

w(b)

and

w^{\prime}(b)

. ►

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

…

6: 1.6 Vectors and Vector-Valued Functions

… ►are tangent to the surface at

\boldsymbol{{\Phi}}(u_{0},v_{0})

. The surface is smooth at this point if

\mathbf{T}_{u}\times\mathbf{T}_{v}\not=0

. …The vector

\mathbf{T}_{u}\times\mathbf{T}_{v}

(u_{0},v_{0})

is normal to the surface at

\boldsymbol{{\Phi}}(u_{0},v_{0})

. … ►A parametrization

\boldsymbol{{\Phi}}(u,v)

of an oriented surface

S

is orientation preserving if

\mathbf{T}_{u}\times\mathbf{T}_{v}

has the same direction as the chosen normal at each point of

S

, otherwise it is orientation reversing. … ►

… ►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … ►In referring to the numerical tables and approximations we use notation typified by

x=0(.05)1

, 8D or 8S. …05, and the corresponding function values are tabulated to 8 decimal places or 8 significant figures. … ►In the Handbook this information is grouped at the section level and appears under the heading Sources in the References section. In the DLMF this information is provided in pop-up windows at the subsection level. …

8: 3.2 Linear Algebra

… ►In solving

\mathbf{A}\mathbf{x}=[1,1,1]^{\rm T}

, we obtain by forward elimination

\mathbf{y}=[1,-1,3]^{\rm T}

, and by back substitution

\mathbf{x}=[\frac{1}{6},\frac{1}{6},\frac{1}{6}]^{\rm T}

. … ►where

\rho(\mathbf{A}\mathbf{A}^{\rm T})

is the largest of the absolute values of the eigenvalues of the matrix

\mathbf{A}\mathbf{A}^{\rm T}

; see §3.2(iv). … ►A nonzero vector

\mathbf{y}

is called a left eigenvector of

\mathbf{A}

corresponding to the eigenvalue

\lambda

\mathbf{y}^{\rm T}\mathbf{A}=\lambda\mathbf{y}^{\rm T}

or, equivalently,

\mathbf{A}^{\rm T}\mathbf{y}=\lambda\mathbf{y}

. … ►Define the Lanczos vectors

\mathbf{v}_{j}

and coefficients

\alpha_{j}

and

\beta_{j}

\mathbf{v}_{0}=\boldsymbol{{0}}

, a normalized vector

\mathbf{v}_{1}

(perhaps chosen randomly),

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

, and for

j=1,2,\ldots,n-1

by the recursive scheme … ►Start with

\mathbf{v}_{0}=\boldsymbol{{0}}

, vector

\mathbf{v}_{1}

such that

\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

. …

9: 23.20 Mathematical Applications

… ►The curve

C

is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element

o=(0,1,0)

as the point at infinity, the negative of

P=(x,y)

-P=(x,-y)

, and generally

P_{1}+P_{2}+P_{3}=0

on the curve iff the points

P_{1}

P_{2}

P_{3}

are collinear. … ►Let

T

denote the set of points on

C

that are of finite order (that is, those points

P

for which there exists a positive integer

n

with

nP=o

), and let

I, K

be the sets of points with integer and rational coordinates, respectively. Then

\emptyset\subseteq T\subseteq I\subseteq K\subseteq C

. …Both

T

and

I

are finite sets. …Values of

x

are then found as integer solutions of

x^{3}+ax+b-y^{2}=0

(in particular

x

must be a divisor of

b-y^{2}

). …

10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►The operator

T

is called self-adjoint if

{T}^{*}=T

, and referred to as symmetric if (1.18.23) holds for

v, w

in the dense domain

\mathcal{D}(T)

T

. … ►If

T\subset T^{**}={T}^{*}

then

T

is essentially self-adjoint and if

T={T}^{*}

then

T

is self-adjoint. … ►Such an operator

T

is called injective if, for any

u, v

in its domain,

Tu=Tv

implies that

u=v

. … ►

The point spectrum $\boldsymbol{\sigma}_{p}$ . It consists of all $z\in\mathbb{C}$ for which $z-T$ is not injective, or equivalently, for which $z$ is an eigenvalue of $T$ , i.e., $Tv=zv$ for some $v\in\mathcal{D}(T)\backslash\{0\}$ .

… ►If

n_{1}=1

then there are no nonzero boundary values at

a

; if

n_{1}=2

then the above boundary values at

a

form a two-dimensional class. …

value at T=0

1—10 of 14 matching pages

1: 35.8 Generalized Hypergeometric Functions of Matrix Argument

Value at $\mathbf{T}=\boldsymbol{{0}}$

2: 4.13 Lambert $W$ -Function

3: 3.11 Approximation Techniques

4: 21.5 Modular Transformations

5: 3.7 Ordinary Differential Equations

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

6: 1.6 Vectors and Vector-Valued Functions

7: Mathematical Introduction

8: 3.2 Linear Algebra

9: 23.20 Mathematical Applications

10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

value at T=0

1—10 of 14 matching pages

1: 35.8 Generalized Hypergeometric Functions of Matrix Argument

Value at 𝐓 = 𝟎

2: 4.13 Lambert W -Function

3: 3.11 Approximation Techniques

4: 21.5 Modular Transformations

5: 3.7 Ordinary Differential Equations

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

6: 1.6 Vectors and Vector-Valued Functions

7: Mathematical Introduction

8: 3.2 Linear Algebra

9: 23.20 Mathematical Applications

10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

Value at $\mathbf{T}=\boldsymbol{{0}}$

2: 4.13 Lambert $W$ -Function