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1: 6.16 Mathematical Applications
§6.16(i) The Gibbs Phenomenon
This nonuniformity of convergence is an illustration of the Gibbs phenomenon. …
See accompanying text
Figure 6.16.1: Graph of S n ( x ) , n = 250 , 0.1 x 0.1 , illustrating the Gibbs phenomenon. Magnify
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
2: 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 ζ ( s , a ) . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • 3: Bibliography G
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • 4: 20 Theta Functions
    Chapter 20 Theta Functions
    5: 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. …
    6: Bibliography I
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • A. R. Its and A. A. Kapaev (2003) Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.
  • 7: 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 example, using double precision d 20 is found to agree with (2.11.31) to 13D. …
    8: 9.10 Integrals
    9.10.14 0 e p t Ai ( t ) d t = e p 3 / 3 ( 1 3 p F 1 1 ( 1 3 ; 4 3 ; 1 3 p 3 ) 3 4 / 3 Γ ( 4 3 ) + p 2 F 1 1 ( 2 3 ; 5 3 ; 1 3 p 3 ) 3 5 / 3 Γ ( 5 3 ) ) , p .
    9.10.15 0 e p t Ai ( t ) d t = 1 3 e p 3 / 3 ( Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) + Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) ) , p > 0 ,
    9.10.16 0 e p t Bi ( t ) d t = 1 3 e p 3 / 3 ( Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) ) , p > 0 .
    9: Bibliography K
  • A. A. Kapaev (1991) Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187, pp. 139–170 (Russian).
  • A. A. Kapaev (2004) Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A 37 (46), pp. 11149–11167.
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • 10: 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). …