# of closed contour

(0.001 seconds)

## 1—10 of 16 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
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 …
###### Winding Number
If $C$ is a closed contour, and $z_{0}\not\in C$, then …
##### 3: 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
1.10.8 $\frac{1}{2\pi\mathrm{i}}\int_{C}f(z)\,\mathrm{d}z=\mbox{sum of the residues of% f(z) within C}.$
1.10.10 $\frac{1}{2\pi\mathrm{i}}\int_{C}\frac{zf^{\prime}(z)}{f(z)}\,\mathrm{d}z=\mbox% {(sum of locations of zeros)}-\mbox{(sum of locations of poles)},$
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 $C$. …
##### 4: 18.10 Integral Representations
18.10.8 $p_{n}(x)=\frac{g_{0}(x)}{2\pi\mathrm{i}}\int_{C}\left(g_{1}(z,x)\right)^{n}g_{% 2}(z,x)(z-c)^{-1}\,\mathrm{d}z$
Here $C$ is a simple closed contour encircling $z=c$ once in the positive sense. …
##### 5: 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 $z_{0},z_{1},\dots,z_{n}$. …
##### 6: 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 $C$. …
##### 7: 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 $z=0$. …
##### 8: Mathematical Introduction
 $\mathbb{C}$ complex plane (excluding infinity). … $f(z)$ is continuous at all points of a simple closed contour $C$ in $\mathbb{C}$. …
##### 9: 14.25 Integral Representations
where the multivalued functions have their principal values when $1 and are continuous in $\mathbb{C}\setminus(-\infty,1]$. 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).
##### 10: 36.7 Zeros
Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …