congruent points

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1: 36.4 Bifurcation Sets

… ►

§36.4(i) Formulas

►

Critical Points for Cuspoids

… ►

Critical Points for Umbilics

… ►This is the codimension-one surface in

\mathbf{x}

space where critical points coalesce, satisfying (36.4.1) and … ►This is the codimension-one surface in

\mathbf{x}

space where critical points coalesce, satisfying (36.4.2) and …

2: 22.4 Periods, Poles, and Zeros

… ►The other poles are at congruent points, which is the set of points obtained by making translations by

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►The other poles and zeros are at the congruent points. … ►The set of points

z=mK+niK^{\prime}

m,n\in\mathbb{Z}

, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by

mK+niK^{\prime}

, where again

m,n\in\mathbb{Z}

. …

3: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

4: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number

x

is represented as … … ► … ►

IEEE Standard

… ►

Rounding

…

5: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

►

§36.12(i) General Theory for Cuspoids

… ►Correspondence between the

u_{j}(\mathbf{y})

and the

t_{j}(\mathbf{x})

is established by the order of critical points along the real axis when

\mathbf{y}

and

\mathbf{x}

are such that these critical points are all real, and by continuation when some or all of the critical points are complex. …In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

6: 27.13 Functions

… ►where

\delta_{1}(n)

and

\delta_{3}(n)

are the number of divisors of

n

congruent respectively to 1 and 3 (mod 4), and by equating coefficients in (27.13.5) and (27.13.6) Jacobi deduced that …Hence

r_{2}\left(5\right)=8

because both divisors,

1

and

5

, are congruent to

1\pmod{4}

. …

7: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …

8: 15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►Here

\{a_{1},a_{2}\}

\{b_{1},b_{2}\}

\{c_{1},c_{2}\}

are the exponent pairs at the points

\alpha

\beta

\gamma

, respectively. … ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►These constants can be chosen to map any two sets of three distinct points

\{\alpha,\beta,\gamma\}

and

\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}

onto each other. … ►

15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

…

9: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with

a_{n}\left(q\right)

and

b_{n}\left(q\right)

finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. …The branch points are called the exceptional values, and the other points normal values. … ►For a visualization of the first branch point of

a_{0}\left(\mathrm{i}\hat{q}\right)

and

a_{2}\left(\mathrm{i}\hat{q}\right)

see Figure 28.7.1. …

10: Sidebar 9.SB1: Supernumerary Rainbows

… ►Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …