# Stokes sets

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## 1—10 of 12 matching pages

##### 2: Bibliography W
• F. J. Wright (1980) The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.
• ##### 3: 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 $\int gf^{-1/2}\,\mathrm{d}x$ would diverge as $x\to+\infty$. Instead set $f=x+\ln x$, $g=0$. …
##### 4: 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 $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\geq 0$. 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 $\ifrac{\mathrm{d}y}{\mathrm{d}x}=\tan\left(\frac{1}{2}\pi t^{2}\right)$, 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
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. …
##### 6: Bibliography S
• I. M. Sheffer (1939) Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.
• P. Spellucci and P. Pulay (1975) Effective calculation of the incomplete gamma function for parameter values $\alpha=(2n+1)/2$, $n=0,\ldots,5$ . Angew. Informatik 17, pp. 30–32.
• K. Srinivasa Rao (1981) Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.
• A. N. Stokes (1980) A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.
• G. Szegö (1950) On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.
• ##### 7: 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 $\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
7.12.6 $R_{n}^{(\mathrm{f})}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{% -\pi z^{2}t/2}t^{2n-(1/2)}}{t^{2}+1}\,\mathrm{d}t,$
7.12.7 $R_{n}^{(\mathrm{g})}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{% -\pi z^{2}t/2}t^{2n+(1/2)}}{t^{2}+1}\,\mathrm{d}t.$
##### 8: 3.10 Continued Fractions
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, … A more stable version of the algorithm is discussed in Stokes (1980). … 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. …
$D_{1}=1/b_{1},$
The recurrences are continued until $(\nabla C_{n})/C_{n}$ is within a prescribed relative precision. …
##### 9: 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). …
##### 10: 6.12 Asymptotic Expansions
For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)2.11(iv), with $p=1$. …