# Stokes sets

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## 1—10 of 12 matching pages

##### 1: 36.5 Stokes Sets

##### 2: Bibliography W

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The Stokes set of the cusp diffraction catastrophe.
J. Phys. A 13 (9), pp. 2913–2928.
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##### 3: 2.7 Differential Equations

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►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).
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►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 $g=\mathrm{ln}x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty}$. Instead set $f=x+\mathrm{ln}x$, $g=0$. …##### 4: 7.20 Mathematical Applications

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►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).
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►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 $dy/dx=\mathrm{tan}\left(\frac{1}{2}\pi {t}^{2}\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi {t}^{2}$. …##### 5: 1.6 Vectors and Vector-Valued Functions

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►Note: The terminology

*open*and*closed sets*and*boundary points*in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►with $(u,v)\in D$, an open set in the plane. … ►###### Stokes’s Theorem

…##### 6: Bibliography S

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Some properties of polynomial sets of type zero.
Duke Math. J. 5, pp. 590–622.
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Effective calculation of the incomplete gamma function for parameter values $\alpha =(2n+1)/2$, $n=0,\mathrm{\dots},5$
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Angew. Informatik 17, pp. 30–32.
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Computation of angular momentum coefficients using sets of generalized hypergeometric functions.
Comput. Phys. Comm. 22 (2-3), pp. 297–302.
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A stable quotient-difference algorithm.
Math. Comp. 34 (150), pp. 515–519.
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On certain special sets of orthogonal polynomials.
Proc. Amer. Math. Soc. 1, pp. 731–737.
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##### 7: 7.12 Asymptotic Expansions

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►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).
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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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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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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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##### 8: 3.10 Continued Fractions

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►can be converted into a continued fraction $C$ of type (3.10.1), and with the property that the $n$th convergent ${C}_{n}={A}_{n}/{B}_{n}$ to $C$ is equal to the $n$th partial sum of the series in (3.10.3), that is,
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►A more stable version of the algorithm is discussed in Stokes (1980).
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►This forward algorithm achieves efficiency and stability in the computation of the convergents ${C}_{n}={A}_{n}/{B}_{n}$, and is related to the forward series recurrence algorithm.
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${D}_{1}=1/{b}_{1},$

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►The recurrences are continued until $(\nabla {C}_{n})/{C}_{n}$ is within a prescribed relative precision.
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##### 9: 2.11 Remainder Terms; Stokes Phenomenon

###### §2.11 Remainder Terms; Stokes Phenomenon

… ►###### §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). …