# simple closed

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## 1—10 of 25 matching pages

##### 1: 31.6 Path-Multiplicative Solutions

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►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the $z$-plane that encircles ${s}_{1}$ and ${s}_{2}$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$.
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##### 2: 1.9 Calculus of a Complex Variable

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►A

*simple closed contour*is a simple contour, except that $z(a)=z(b)$. … ►Any simple closed contour $C$ divides $\u2102$ into two open domains that have $C$ as common boundary. … ►If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$, then … ►If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$, and if ${z}_{0}$ is a point within $C$, then …##### 3: 1.6 Vectors and Vector-Valued Functions

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►The geometrical image $C$ of a path $\mathbf{c}$ is called a

*simple closed curve*if $\mathbf{c}$ is one-to-one, with the exception $\mathbf{c}(a)=\mathbf{c}(b)$. … … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. …##### 4: 3.3 Interpolation

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3.3.6
$${R}_{n}(z)=\frac{{\omega}_{n+1}(z)}{2\pi \mathrm{i}}{\int}_{C}\frac{f(\zeta )}{(\zeta -z){\omega}_{n+1}(\zeta )}d\zeta ,$$

►where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,{z}_{1},{z}_{2},\mathrm{\dots},{z}_{n}$.
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3.3.37
$$\left[{z}_{0},{z}_{1},\mathrm{\dots},{z}_{n}\right]f=\frac{1}{2\pi \mathrm{i}}{\int}_{C}\frac{f(\zeta )}{{\omega}_{n+1}(\zeta )}d\zeta ,$$

►where ${\omega}_{n+1}(\zeta )$ is given by (3.3.3), and $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing ${z}_{0},{z}_{1},\mathrm{\dots},{z}_{n}$.
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##### 5: 3.4 Differentiation

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3.4.17
$$\frac{1}{k!}{f}^{(k)}({x}_{0})=\frac{1}{2\pi \mathrm{i}}{\int}_{C}\frac{f(\zeta )}{{(\zeta -{x}_{0})}^{k+1}}d\zeta ,$$

►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and ${x}_{0}$ is interior to $C$.
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##### 6: 2.10 Sums and Sequences

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2.10.26
$${f}_{n}=\frac{1}{2\pi \mathrm{i}}{\int}_{\mathcal{C}}\frac{f(z)}{{z}^{n+1}}dz,$$

►where $\mathcal{C}$ is a simple closed contour in the annulus that encloses $z=0$.
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##### 7: 1.10 Functions of a Complex Variable

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►Let $C$ be a simple closed contour consisting of a segment $\mathrm{\mathit{A}\mathit{B}}$ of the real axis and a contour in the upper half-plane joining the ends of $\mathrm{\mathit{A}\mathit{B}}$.
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►If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then
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►If $f(z)$ and $g(z)$ are analytic on and inside a simple closed contour $C$, and $$ on $C$, then $f(z)$ and $f(z)+g(z)$ have the same number of zeros inside $C$.
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##### 8: Mathematical Introduction

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$\u2102$ | complex plane (excluding infinity). |
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${f(z)|}_{C}=0$ | $f(z)$ is continuous at all points of a simple closed contour $C$ in $\u2102$. |

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##### 9: 18.10 Integral Representations

##### 10: 1.4 Calculus of One Variable

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►For an example, see Figure 1.4.1
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►Suppose $f(x)$ is defined on $[a,b]$.
…Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist.
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►If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then
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►A similar definition applies to closed intervals $[a,b]$.
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