# simple closed contour

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## 9 matching pages

##### 1: 31.6 Path-Multiplicative Solutions
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 ${\mathrm{e}}^{2\nu\pi i}$. …
##### 2: 1.9 Calculus of a Complex Variable
A simple closed contour is a simple contour, except that $z(a)=z(b)$. … Any simple closed contour $C$ divides $\mathbb{C}$ 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: 3.3 Interpolation
3.3.6 $R_{n}(z)=\frac{\omega_{n+1}(z)}{2\pi\mathrm{i}}\int_{C}\frac{f(\zeta)}{(\zeta-% z)\omega_{n+1}(\zeta)}\,\mathrm{d}\zeta,$
where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_{1},z_{2},\dots,z_{n}$. …
3.3.37 $\left[z_{0},z_{1},\dots,z_{n}\right]f=\frac{1}{2\pi\mathrm{i}}\int_{C}\frac{f(% \zeta)}{\omega_{n+1}(\zeta)}\,\mathrm{d}\zeta,$
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 $z_{0},z_{1},\dots,z_{n}$. …
##### 4: 3.4 Differentiation
3.4.17 $\frac{1}{k!}\,f^{(k)}(x_{0})=\frac{1}{2\pi i}\int_{C}\frac{f(\zeta)}{(\zeta-x_% {0})^{k+1}}\,\,\mathrm{d}\zeta,$
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 $C$. …
##### 5: 2.10 Sums and Sequences
2.10.26 $f_{n}=\frac{1}{2\pi i}\int_{\mathscr{C}}\frac{f(z)}{z^{n+1}}\,\mathrm{d}z,$
where $\mathscr{C}$ is a simple closed contour in the annulus that encloses $z=0$. …
##### 6: 1.10 Functions of a Complex Variable
Let $C$ be a simple closed contour consisting of a segment $\mathit{AB}$ of the real axis and a contour in the upper half-plane joining the ends of $\mathit{AB}$. … 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 … If $f(z)$ and $g(z)$ are analytic on and inside a simple closed contour $C$, and $\left|g(z)\right|<\left|f(z)\right|$ on $C$, then $f(z)$ and $f(z)+g(z)$ have the same number of zeros inside $C$. …
##### 7: Mathematical Introduction
 $\mathbb{C}$ complex plane (excluding infinity). … $f(z)$ is continuous at all points of a simple closed contour $C$ in $\mathbb{C}$. …
##### 8: 18.10 Integral Representations
Here $C$ is a simple closed contour encircling $z=c$ once in the positive sense. …
##### 9: 2.4 Contour Integrals
###### §2.4 Contour Integrals
is seen to converge absolutely at each limit, and be independent of $\sigma\in[c,\infty)$. … The most successful results are obtained on moving the integration contour as far to the left as possible. … Let $\mathscr{P}$ denote the path for the contour integral … and assigning an appropriate value to $c$ to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …