# on a region

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## 1—10 of 60 matching pages

##### 1: 21.10 Methods of Computation

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##### 2: 1.9 Calculus of a Complex Variable

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*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 $\u03f5$ ($>0$) we can find a neighborhood of ${z}_{0}$ such that $$ for all points $z$ in the intersection of the neighborhood with $R$. …##### 3: 2.4 Contour Integrals

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►If $q(t)$ is analytic in a sector $$ containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths.
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►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).
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##### 4: 28.17 Stability as $x\to \pm \mathrm{\infty}$

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►For real $a$ and $q$
$(\ne 0)$ 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

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►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty}$ and $x=0$, respectively.
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##### 6: 10.72 Mathematical Applications

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►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 $\frac{1}{3}$ (§9.6(i)).
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►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 $z\to {z}_{0}$.
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##### 7: 19.20 Special Cases

##### 8: Bibliography M

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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.
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##### 9: 1.5 Calculus of Two or More Variables

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►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
$${f}^{\ast}(x,y)=\{\begin{array}{cc}f(x,y),\hfill & \text{if}(x,y)\in D,\hfill \\ 0,\hfill & \text{if}(x,y)\in R\setminus D\text{.}\hfill \end{array}$$

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1.5.29
$${\iint}_{D}f(x,y)dA={\iint}_{R}{f}^{\ast}(x,y)dA,$$

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1.5.31
$${\iint}_{D}f(x,y)dA={\int}_{a}^{b}{\int}_{{\varphi}_{1}(x)}^{{\varphi}_{2}(x)}f(x,y)dydx,$$

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##### 10: 13.9 Zeros

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►Then $M(a,b,z)$ has no zeros in the regions
${P}_{b/a}$, if $$; ${P}_{1}$, if $1\le a\le b$; ${P}_{\alpha}$, where $\alpha =(2a-b+ab)/(a(a+1))$, if $$ and $$.
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