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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 ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. …

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 $z_0\notin C$, then …

3: 1.10 Functions of a Complex Variable

… ►Let $C$ be a simple closed contour consisting of a segment $\mathrm{𝐴𝐵}$ of the real axis and a contour in the upper half-plane joining the ends of $\mathrm{𝐴𝐵}$. … ►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 $$\frac{1}{2\pi \mathrm{i}}{\int }_{C}f(z)dz=\text{sum of the residues of }f(z)\text{ within }C.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: variable and $C$: closed contour

Referenced by:

§2.10(iii)

Permalink:

http://dlmf.nist.gov/1.10.E8

Encodings:

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Annotations for §1.10(iv), §1.10 and Ch.1

… ►

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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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: variable and $C$: closed contour

Permalink:

http://dlmf.nist.gov/1.10.E10

Encodings:

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Annotations for §1.10(iv), §1.10(iv), §1.10 and Ch.1

… ►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 $C$. …

4: 18.10 Integral Representations

… ►

18.10.8 $$p_n(x)=\frac{g_0(x)}{2\pi \mathrm{i}}{\int }_C{\left(g_1(z,x)\right)}^n{g}_2(z,x){(z-c)}^{-1}dz$$

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Defines:

$g_0(x)$: factors (locally), $g_1(z,x)$: factors (locally), $g_2(z,x)$: factor (locally) and $c$: center (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $C$: closed contour and $x$: real variable

Referenced by:

Table 18.10.1, (18.17.21_1)

Permalink:

http://dlmf.nist.gov/18.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iii), §18.10 and Ch.18

… ►Here $C$ is a simple closed contour encircling $z=c$ once in the positive sense. …

5: 3.3 Interpolation

… ►

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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, ${\omega }_{n+1}(z)$: nodal polynomial, $R_n(x)$: remainder and $C$: simple closed contour in $D$

A&S Ref:

25.2.5

Permalink:

http://dlmf.nist.gov/3.3.E6

Encodings:

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Annotations for §3.3(i), §3.3 and Ch.3

►where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_1,z_2,\mathrm{\dots },z_n$. … ►

3.3.37 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{\omega }_{n+1}(\zeta )}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, $C$: simple closed contour in $D$ , $z_k$: nodes of function and ${\omega }_{n+1}(z)$: nodal polynomial

Permalink:

http://dlmf.nist.gov/3.3.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

►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 $z_0,z_1,\mathrm{\dots },z_n$. …

6: 3.4 Differentiation

… ►

3.4.17 $$\frac{1}{k!}{f}^{(k)}(x_0)=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{(\zeta -x_0)}^{k+1}}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $C$: simple closed contour

Permalink:

http://dlmf.nist.gov/3.4.E17

Encodings:

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Annotations for §3.4(ii), §3.4 and Ch.3

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

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(x)$: analytic function, $f_n$: coefficients and $𝒞$: simple closed contour

Permalink:

http://dlmf.nist.gov/2.10.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

►where $𝒞$ is a simple closed contour in the annulus that encloses $z=0$. …

8: Mathematical Introduction

… ► ►►►

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

…

9: 14.25 Integral Representations

… ►where the multivalued functions have their principal values when and are continuous in $ℂ\setminus (-\mathrm{\infty },1]$. ►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).

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 …