# Gibbs phenomenon

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## 1—10 of 16 matching pages

##### 1: 6.16 Mathematical Applications
###### §6.16(i) The GibbsPhenomenon
This nonuniformity of convergence is an illustration of the Gibbs phenomenon. … Figure 6.16.1: Graph of S n ⁡ ( x ) , n = 250 , − 0.1 ≤ x ≤ 0.1 , illustrating the Gibbs phenomenon. Magnify
##### 2: 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. …
##### 3: 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). …
##### 4: 8.22 Mathematical Applications
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. …
##### 5: Bibliography P
• R. B. Paris and A. D. Wood (1995) Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.
• R. B. Paris (1992a) Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.
• R. B. Paris (1992b) Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
• R. B. Paris (2005b) The Stokes phenomenon associated with the Hurwitz zeta function $\zeta(s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.
• ##### 6: 9.10 Integrals
9.10.14 $\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=e^{-p^{3}% /3}\left(\frac{1}{3}-\frac{p{{}_{1}F_{1}}\left(\tfrac{1}{3};\tfrac{4}{3};% \tfrac{1}{3}p^{3}\right)}{3^{4/3}\Gamma\left(\tfrac{4}{3}\right)}+\frac{p^{2}{% {}_{1}F_{1}}\left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3^{5/3}% \Gamma\left(\tfrac{5}{3}\right)}\right),$ $p\in\mathbb{C}$.
9.10.15 $\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(-t\right)\,\mathrm{d}t={\frac{1% }{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{% \Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^% {3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},$ $\Re p>0$,
9.10.16 $\int_{0}^{\infty}e^{-pt}\operatorname{Bi}\left(-t\right)\,\mathrm{d}t={\frac{1% }{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}% \right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},% \tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)},$ $\Re p>0$.
##### 7: 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? …
##### 8: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when $\rho>0$, and Wong and Zhao (1999a) when $-1<\rho<0$. …
##### 9: Bibliography I
• A. R. Its and A. A. Kapaev (2003) Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.
• ##### 10: 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). …