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1: Sidebar 9.SB2: Interference Patterns in Caustics

… ►The bright sharp-edged triangle is a caustic, that is a line of focused light. …

2: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The faint circular patterns are additional diffraction effects due to imperfections in the edge.

3: About MathML

… ►Most current browsers based on this engine now support MathML natively, including Chrome, Microsoft Edge, and Safari. …

4: 36.5 Stokes Sets

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. …

5: 7.3 Graphics

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. Fresnel (1818) introduced the integral $ℱ\left(x\right)$ in his study of the interference pattern at the edge of a shadow. … Magnify

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6: 16.23 Mathematical Applications

… ►A substantial transition occurs in a random graph of $n$ vertices when the number of edges becomes approximately $\frac{1}{2}n$. …

7: 23.20 Mathematical Applications

… ►The two pairs of edges $[0,{\omega }_1]\cup [{\omega }_1,2{\omega }_3]$ and $[2{\omega }_3,2{\omega }_3-{\omega }_1]\cup [2{\omega }_3-{\omega }_1,0]$ of $R$ are each mapped strictly monotonically by $\mathrm{\wp }$ onto the real line, with $0\to \mathrm{\infty }$, ${\omega }_1\to e_1$, $2{\omega }_3\to -\mathrm{\infty }$; similarly for the other pair of edges. For each pair of edges there is a unique point $z_0$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►The cut is the image of the edge from $\frac{1}{2}{\omega }_1+{\omega }_3$ to $\frac{1}{2}{\omega }_1-{\omega }_3$ and is not a line segment. …

8: 34.4 Definition: $\mathit{6}j$ Symbol

… ►Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3}j$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_1,j_2,j_3,l_1,l_2,l_3$; see Figure 34.4.1. …

9: 19.3 Graphics

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2\ge 1$) it has the value $K\left(k^{\prime }\right)$ if $k^2>1$, and $\frac{1}{4}\pi$ if $k^2=1$. Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2>1$) it has the (negative) value $K\left(k^{\prime }\right)-E\left(k^{\prime }\right)$, with limit 0 as $k^2\to 1+$. Magnify 3D Help

10: Bibliography J

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Introduction to Asymptotics: A Treatment Using Nonstandard Analysis. World Scientific Publishing Co. Inc., River Edge, NJ.

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External Links:

ISBN 981-02-2915-1, MathReview (W. A. J. Luxemburg), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

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