# on a region

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##### 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
A 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. … A function $f(z)$ is continuous on a region $R$ if for each point $z_{0}$ in $R$ 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
If $q(t)$ 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. … The 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 $x\to\pm\infty$
For 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
Luke (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
In 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)). … In 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 $(z-z_{0})^{2}g(z)$ as $z\to z_{0}$. …
##### 7: 19.20 Special Cases
Since $x, $p$ is in a hyperbolic region. …
##### 8: Bibliography M
• 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.
• ##### 9: 1.5 Calculus of Two or More Variables
A function is continuous on a point set $D$ if it is continuous at all points of $D$. … For $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let
1.5.28 $f^{*}(x,y)=\begin{cases}f(x,y),&\mbox{if (x,y)\in D},\\ 0,&\mbox{if (x,y)\in R\setminus D.}\end{cases}$
1.5.31 $\iint_{D}f(x,y)\,\mathrm{d}A=\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y% )\,\mathrm{d}y\,\mathrm{d}x,$
##### 10: 13.9 Zeros
Then $M\left(a,b,z\right)$ has no zeros in the regions $P_{\ifrac{b}{a}}$, if $0; $P_{1}$, if $1\leq a\leq b$; $P_{\alpha}$, where $\alpha=\ifrac{(2a-b+ab)}{(a(a+1))}$, if $0 and $a\leq b<\ifrac{2a}{(1-a)}$. …