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21: 36.5 Stokes Sets

… ►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …

22: 1.6 Vectors and Vector-Valued Functions

… ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►

1.6.44 $${\iint }_S\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA={\int }_C𝐅\cdot d𝐬={\int }_C{F}_{1}dx+F_{2}dy.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝐅(x,y)$: vector, $S$: closed region, $C$: closed curve and $F_j$: vector function components

Referenced by:

§1.6(iv)

Permalink:

http://dlmf.nist.gov/1.6.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iv), §1.6(iv), §1.6 and Ch.1

… ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. … ►

1.6.58 $${\iiint }_V(\nabla \cdot 𝐅)dV={\iint }_S𝐅\cdot d𝐒,$$

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Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

… ►

1.6.59 $${\iiint }_V(f{\nabla }^{2}g+\nabla f\cdot \nabla g)dV={\iint }_{S}f\frac{\partial g}{\partial n}dA,$$

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Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E59

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

…

23: 1.5 Calculus of Two or More Variables

… ►A function is continuous on a point set $D$ if it is continuous at all points of $D$. … ►For $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let ►

1.5.28

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Symbols:

$\in$: element of, $(a,b)$: open interval, $\setminus$: set subtraction, $R$: closed rectangle and $D$: region contained in $R$

Permalink:

http://dlmf.nist.gov/1.5.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

… ►

1.5.29 $${\iint }_{D}f(x,y)dA={\iint }_R{f}^{\ast }(x,y)dA,$$

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Symbols:

$dx$: differential of $x$ , $R$: closed rectangle and $D$: region contained in $R$

Permalink:

http://dlmf.nist.gov/1.5.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

… ►

1.5.31 $${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{{\varphi }_1(x)}^{{\varphi }_2(x)}f(x,y)dydx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $D$: region contained in $R$

Permalink:

http://dlmf.nist.gov/1.5.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

…

24: 1.9 Calculus of a Complex Variable

… ►A region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points. ►A function $f(z)$ is continuous on a region $R$ if for each point $z_0$ in $R$ and any given number $ϵ$ ($>0$) we can find a neighborhood of $z_0$ such that for all points $z$ in the intersection of the neighborhood with $R$. …

25: Bibliography W

… ►

Asymptotic behavior of stability regions for Hill’s equation. SIAM J. Appl. Math. 47 (5), pp. 941–958.

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External Links:

ISSN 0036-1399, Document, MathReview (Ekkehard Wagenführer), ZentralBlatt

Referenced by:

§28.29(iii).

Encodings:

BibTeX

… ►

Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region. Phys. Rev. B 13, pp. 316–374.

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External Links:

Document, Link

Referenced by:

§32.16, §32.16.

Encodings:

BibTeX

…

26: 10.2 Definitions

… ►Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\mathrm{\Re }\nu \ge 0$. … ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

► ►►

Pair	Interval or Region
…

►

27: 2.4 Contour Integrals

… ►If $q(t)$ is analytic in a sector containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. … ►Additionally, it may be advantageous to arrange that $\mathrm{\Im }\left(zp(t)\right)$ is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. … ►The problem of obtaining an asymptotic approximation to $I(\alpha ,z)$ that is uniform with respect to $\alpha$ in a region containing $\widehat{\alpha }$ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). …

28: 10.25 Definitions

… ►

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.

► ►►

Pair	Region
…

►

29: 19.33 Triaxial Ellipsoids

… ►If a conducting ellipsoid with semiaxes $a,b,c$ bears an electric charge $Q$, then the equipotential surfaces in the exterior region are confocal ellipsoids: …

30: 28.33 Physical Applications

… ►Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as $(q,a)$ is in the intersection of $ℒ$ with the colored or the uncolored open regions depicted in Figure 28.17.1. …