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1: 36.5 Stokes Sets

§36.5 Stokes Sets

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§36.5(i) Definitions

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§36.5(ii) Cuspoids

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Elliptic Umbilic Stokes Set (Codimension three)

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§36.5(iv) Visualizations

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2: 36.3 Visualizations of Canonical Integrals

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Figure 36.3.1: Modulus of Pearcey integral $|{\mathrm{\Psi }}_2\left(x,y\right)|$.

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Figure 36.3.2: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,3\right)|$.

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Figure 36.3.3: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,0\right)|$.

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Figure 36.3.4: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-3\right)|$.

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Figure 36.3.13: Phase of Pearcey integral $\mathrm{ph}{\mathrm{\Psi }}_2\left(x,y\right)$.

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3: 36.4 Bifurcation Sets

§36.4 Bifurcation Sets

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Bifurcation (Catastrophe) Set for Cuspoids

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Bifurcation (Catastrophe) Set for Umbilics

… ► $K=1$, fold bifurcation set: … ►

§36.4(ii) Visualizations

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4: Sidebar 9.SB1: Supernumerary Rainbows

… ►Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. The house in the picture is Newton’s birthplace. …

5: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\ge 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. …Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. …

6: 2.11 Remainder Terms; Stokes Phenomenon

§2.11 Remainder Terms; Stokes Phenomenon

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§2.11(iv) Stokes Phenomenon

… ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. Where should the change-over take place? Can it be accomplished smoothly? … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). …

7: 2.7 Differential Equations

… ►To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing $z$ in (2.7.1) with $1/z$; see Olver (1997b, pp. 153–154). … ►in which $C_1$, $C_2$ are constants, the so-called Stokes multipliers. … ►For the calculation of Stokes multipliers see Olde Daalhuis and Olver (1995b). … ►We cannot take $f=x$ and $g=\ln x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty }$. Instead set $f=x+\ln x$, $g=0$. …

8: About the Project

… ►These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns. …

9: 7.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). ►For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). … ►

7.12.2 $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}},$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.27

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E2

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.6 $$R_n^{(\mathrm{f})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n-(1/2)}}{t^2+1}dt,$$

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Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.29 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.7 $$R_n^{(\mathrm{g})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n+(1/2)}}{t^2+1}dt.$$

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Defines:

$R_n^{(\mathrm{g})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.30 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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10: Bibliography W

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.

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External Links:

ISSN 0305-4470, Document, MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.5(ii).

Encodings:

BibTeX

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