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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 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: 1.6 Vectors and Vector-Valued Functions

… ►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

… ►

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,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : 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:: pMML, png, TeX
See also:: 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},\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,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$ : divided difference, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $f(z)$ : function, $C$ : simple closed contour in $D$ , $z_{\NVar{k}}$ : 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},\dots,z_{n}

. …

5: 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,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : 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:: pMML, png, TeX
See also:: 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

. …

6: 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,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $f(x)$ : analytic function, $f_{n}$ : coefficients and $\mathscr{C}$ : 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

\mathscr{C}

is a simple closed contour in the annulus that encloses

z=0

. …

7: 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|

C

, then

f(z)

and

f(z)+g(z)

have the same number of zeros inside

C

. …

8: Mathematical Introduction

… ► ►►►

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

…

9: 18.10 Integral Representations

… ►Here

C

is a simple closed contour encircling

z=c

once in the positive sense. …

10: 1.4 Calculus of One Variable

… ►For an example, see Figure 1.4.1 … ►Suppose

f(x)

is defined on

[a,b]

. …Continuity, or piecewise continuity, of

f(x)

[a,b]

is sufficient for the limit to exist. … ►If

f(x)

is continuous or piecewise continuous on

[a,b]

, then … ►A similar definition applies to closed intervals

[a,b]

. …