on a region

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1—10 of 60 matching pages

1: 21.10 Methods of Computation

•

Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

…

2: 1.9 Calculus of a Complex Variable

… в–єA region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points. … в–єA function

f(z)

is continuous on a region

R

if for each point

z_{0}

R

and any given number

\epsilon

(

>0

) we can find a neighborhood of

z_{0}

such that

\left|f(z)-f(z_{0})\right|<\epsilon

for all points

z

in the intersection of the neighborhood with

R

. …

3: 2.4 Contour Integrals

… в–єIf

q(t)

is analytic in a sector

\alpha_{1}<\operatorname{ph}t<\alpha_{2}

containing

\operatorname{ph}t=0

, then the region of validity may be increased by rotation of the integration paths. … в–єThe problem of obtaining an asymptotic approximation to

I(\alpha,z)

that is uniform with respect to

\alpha

in a region containing

\widehat{\alpha}

is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). …

4: 28.17 Stability as $x\to\pm\infty$

… в–єFor real

a

and

q

(\neq 0)

the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves

a=a_{n}\left(q\right)

and

a=b_{n}\left(q\right)

; compare Figure 28.2.1. …

5: 12.20 Approximations

… в–єLuke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions

U\left(a,b,x\right)

and

M\left(a,b,x\right)

(§13.2(i)) whose regions of validity include intervals with endpoints

x=\infty

and

x=0

, respectively. …

6: 10.72 Mathematical Applications

… в–єIn regions in which (10.72.1) has a simple turning point

z_{0}

, that is,

f(z)

and

g(z)

are analytic (or with weaker conditions if

z=x

is a real variable) and

z_{0}

is a simple zero of

f(z)

, asymptotic expansions of the solutions

w

for large

u

can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order

\tfrac{1}{3}

(§9.6(i)). … в–єIn regions in which the function

f(z)

has a simple pole at

z=z_{0}

and

(z-z_{0})^{2}g(z)

is analytic at

z=z_{0}

(the case

\lambda=-1

in §10.72(i)), asymptotic expansions of the solutions

w

of (10.72.1) for large

u

can be constructed in terms of Bessel functions and modified Bessel functions of order

\pm\sqrt{1+4\rho}

, where

\rho

is the limiting value of

(z-z_{0})^{2}g(z)

z\to z_{0}

. …

7: 19.20 Special Cases

… в–єSince

x<y <mrow> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>z</mi> </mrow> </math>, <math class=

is in a hyperbolic region. …

8: Bibliography M

… в–є

A. Michaeli (1996) Asymptotic analysis of edge-excited currents on a convex face of a perfectly conducting wedge under overlapping penumbra region conditions. IEEE Trans. Antennas and Propagation 44 (1), pp. 97–101.

в“

External Links:: Document
Referenced by:: §9.14.
Encodings:: BibTeX

…

9: 1.5 Calculus of Two or More Variables

… в–єA function is continuous on a point set

D

if it is continuous at all points of

D

. … в–єFor

f(x,y)

defined on a point set

D

contained in a rectangle

R

, let в–є

1.5.28

f^{*}(x,y)=\begin{cases}f(x,y),&\mbox{if $(x,y)\in D$},\\ 0,&\mbox{if $(x,y)\in R\setminus D$.}\end{cases}

в“

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $R$ : closed rectangle and $D$ : region contained in $R$
Permalink:: http://dlmf.nist.gov/1.5.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

… в–є

1.5.29

\iint_{D}f(x,y)\,\mathrm{d}A=\iint_{R}f^{*}(x,y)\,\mathrm{d}A,

в“

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R$ : closed rectangle and $D$ : region contained in $R$
Permalink:: http://dlmf.nist.gov/1.5.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

… в–є

1.5.31

\iint_{D}f(x,y)\,\mathrm{d}A=\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y% )\,\mathrm{d}y\,\mathrm{d}x,

в“

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $D$ : region contained in $R$
Permalink:: http://dlmf.nist.gov/1.5.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

…

10: 13.9 Zeros

… в–єThen

M\left(a,b,z\right)

has no zeros in the regions

P_{\ifrac{b}{a}}

, if

0<b\leq a

;

P_{1}

, if

1\leq a\leq b

;

P_{\alpha}

, where

\alpha=\ifrac{(2a-b+ab)}{(a(a+1))}

, if

0<a<1

and

a\leq b<\ifrac{2a}{(1-a)}

. …

on a region

1—10 of 60 matching pages

1: 21.10 Methods of Computation

2: 1.9 Calculus of a Complex Variable

3: 2.4 Contour Integrals

4: 28.17 Stability as x → ± ∞

5: 12.20 Approximations

6: 10.72 Mathematical Applications

7: 19.20 Special Cases

8: Bibliography M

9: 1.5 Calculus of Two or More Variables

10: 13.9 Zeros

4: 28.17 Stability as $x\to\pm\infty$