interior points

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8 matching pages

1: 1.9 Calculus of a Complex Variable

… ►Points of a region that are not boundary points are called interior points. … ►

Jordan Curve Theorem

…

3: 19.2 Definitions

… ►The integral for

E\left(\phi,k\right)

is well defined if

k^{2}={\sin}^{2}\phi=1

, and the Cauchy principal value (§1.4(v)) of

\Pi\left(\phi,\alpha^{2},k\right)

is taken if

1-\alpha^{2}{\sin}^{2}\phi

vanishes at an interior point of the integration path. …

4: 23.20 Mathematical Applications

… ►The interior of

R

is mapped one-to-one onto the lower half-plane. … ►The interior of the rectangle with vertices

0

\omega_{1}

2\omega_{3}

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

is mapped two-to-one onto the lower half-plane. … ►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. …The resulting points are then tested for finite order as follows. …If any of these quantities is zero, then the point has finite order. …

5: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With

e

denoting here the elementary charge, the Coulomb potential between two point particles with charges

Z_{1}e,Z_{2}e

and masses

m_{1},m_{2}

separated by a distance

s

V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s

, where

Z_{j}

are atomic numbers,

\varepsilon_{0}

is the electric constant,

\alpha

is the fine structure constant, and

\hbar

is the reduced Planck’s constant. … ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions,

F_{\ell}\left(\eta,\rho\right)

and

G_{\ell}\left(\eta,\rho\right)

, or

f\left(\epsilon,\ell;r\right)

and

h\left(\epsilon,\ell;r\right)

, to determine the scattering

S

-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). … ►

§33.22(vi) Solutions Inside the Turning Point

…

6: 1.15 Summability Methods

… ►at every point

\theta

where both limits exist. … ►can be extended to the interior of the unit circle as an analytic function …

7: 3.4 Differentiation

… ►

Two-Point Formula

… ►

Three-Point Formula

… ►

Four-Point Formula

… ►

Five-Point Formula

… ►

Six-Point Formula

…

8: 2.1 Definitions and Elementary Properties

… ►Let

\mathbf{X}

be a point set with a limit point

c

. As

x\to c

\mathbf{X}

… ►If

c

is a finite limit point of

\mathbf{X}

, then … ►Similarly for finite limit point

c

in place of

\infty

. … ►where

c

is a finite, or infinite, limit point of

\mathbf{X}

. …