# of closed contour

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## 1—10 of 16 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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►If $f(z)$ is continuous within and on a simple closed contour
$C$ and analytic within $C$, then
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►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
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###### Winding Number

►If $C$ is a closed contour, and ${z}_{0}\notin C$, then …##### 3: 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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1.10.8
$$\frac{1}{2\pi \mathrm{i}}{\int}_{C}f(z)dz=\text{sum of the residues of}f(z)\text{within}C.$$

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1.10.10
$$\frac{1}{2\pi \mathrm{i}}{\int}_{C}\frac{z{f}^{\prime}(z)}{f(z)}dz=\text{(sum of locations of zeros)}-\text{(sum of locations of poles)},$$

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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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##### 4: 18.10 Integral Representations

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18.10.8
$${p}_{n}(x)=\frac{{g}_{0}(x)}{2\pi \mathrm{i}}{\int}_{C}{\left({g}_{1}(z,x)\right)}^{n}{g}_{2}(z,x){(z-c)}^{-1}dz$$

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►Here $C$ is a simple closed contour encircling $z=c$ once in the positive sense.
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##### 5: 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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##### 6: 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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##### 7: 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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##### 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: 14.25 Integral Representations

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►where the multivalued functions have their principal values when $$ and are continuous in $\u2102\setminus (-\mathrm{\infty},1]$.
►For corresponding contour integrals, with less restrictions on $\mu $ and $\nu $, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).