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11: 31.10 Integral Equations and Representations

… ►for a suitable contour

C

. …The contour

C

must be such that … ►Here

\tilde{\kappa}_{m}

is a normalization constant and

C

is the contour of Example 1. … ►for suitable contours

C_{1}

C_{2}

. …The contours

C_{1}

C_{2}

must be chosen so that …

12: 36.3 Visualizations of Canonical Integrals

… ►In Figure 36.3.13(a) points of confluence of phase contours are zeros of

\Psi_{2}\left(x,y\right)

; similarly for other contour plots in this subsection. … ►

See accompanying text — Figure 36.3.15: Phase of elliptic umbilic canonical integral $\operatorname{ph}\Psi^{(\mathrm{E})}\left(x,y,0\right)$ . … Magnify

►

…

13: 1.9 Calculus of a Complex Variable

… ►

§1.9(iii) Integration

… ►A simple closed contour is a simple contour, except that

z(a)=z(b)

. … ►If

f(z)

is continuous within and on a simple closed contour

C

and analytic within

C

, then … ►

Winding Number

…

14: 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}

. … ►and the integration contour is described once in the positive sense. … ►Each contour is called a cut. …(Or more generally, a simple contour that starts at the center and terminates on the boundary.) … ►

§1.10(viii) Functions Defined by Contour Integrals

…

15: 18.10 Integral Representations

… ►

§18.10(iii) Contour Integral Representations

►Table 18.10.1 gives contour integral representations of the form ►

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

ⓘ

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, $\,\mathrm{d}\NVar{x}$ : 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. ►

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$	$c$	Conditions
…

► …

16: 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}

. …

17: 31.9 Orthogonality

… ►

31.9.2

\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\zeta$ : point and $\theta_{m}$ : normalization constant
Permalink:: http://dlmf.nist.gov/31.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

… ►The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). … ►and the integration paths

\mathcal{L}_{1}

\mathcal{L}_{2}

are Pochhammer double-loop contours encircling distinct pairs of singularities

\{0,1\}

\{0,a\}

\{1,a\}

. …

18: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►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 ►

2.4.10

I(z)=\int_{a}^{b}e^{-zp(t)}q(t)\,\mathrm{d}t,

ⓘ

Defines:: $I(z)$ : contour integral (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $a$ : left endpoint, $b$ : right endpoint, $p(t)$ : analytic function and $q(t)$ : analytic function
Referenced by:: §2.4(iv)
Permalink:: http://dlmf.nist.gov/2.4.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(iii), §2.4 and Ch.2

… ►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). …

19: 13.16 Integral Representations

… ►

§13.16(ii) Contour Integrals

►For contour integral representations combine (13.14.2) and (13.14.3) with §13.4(ii). … ►where the contour of integration separates the poles of

\Gamma\left(t-\kappa\right)

from those of

\Gamma\left(\frac{1}{2}+\mu-t\right)

. … ►where the contour of integration separates the poles of

\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)

from those of

\Gamma\left(-\kappa-t\right)

. …where the contour of integration passes all the poles of

\Gamma\left(\frac{1}{2}+\mu+t\right)\Gamma\left(\frac{1}{2}-\mu+t\right)

on the right-hand side.

20: 21.9 Integrable Equations

… ►

…


(a) Contour plot.	(b) Density plot.


(a) Contour plot.	(b) Density plot.


(a) Contour plot.	(b) Density plot.


(a) Contour plot.	(b) Density plot.