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connection formulas across transition points

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1: 36.4 Bifurcation Sets
§36.4(i) Formulas
K = 2 , cusp bifurcation set: … K = 3 , swallowtail bifurcation set: … Elliptic umbilic bifurcation set (codimension three): for fixed z , the section of the bifurcation set is a three-cusped astroid … Hyperbolic umbilic bifurcation set (codimension three): …
2: 2.8 Differential Equations with a Parameter
§2.8(ii) Case I: No Transition Points
For error bounds, more delicate error estimates, extensions to complex ξ and u , zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … For results, including error bounds, see Olver (1977c). For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).
§2.8(vi) Coalescing Transition Points
3: 36.5 Stokes Sets
where j denotes a real critical point (36.4.1) or (36.4.2), and μ denotes a critical point with complex t or s , t , connected with j by a steepest-descent path (that is, a path where Φ = constant ) in complex t or ( s , t ) space. … The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … This part of the Stokes set connects two complex saddles. … In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …
4: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). … In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. … PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
5: Bibliography O
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
  • 6: Bibliography W
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.
  • G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.
  • R. Wong and H. Y. Zhang (2009a) On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. (Singap.) 7 (4), pp. 419–448.
  • R. Wong and H. Y. Zhang (2009b) On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.
  • 7: 31.18 Methods of Computation
    Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …
    8: 14.21 Definitions and Basic Properties
    P ν ± μ ( z ) and 𝑸 ν μ ( z ) exist for all values of ν , μ , and z , except possibly z = ± 1 and , which are branch points (or poles) of the functions, in general. …
    §14.21(iii) Properties
    This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
    9: 14.9 Connection Formulas
    §14.9 Connection Formulas
    §14.9(i) Connections Between 𝖯 ν ± μ ( x ) , 𝖯 ν 1 ± μ ( x ) , 𝖰 ν ± μ ( x ) , 𝖰 ν 1 μ ( x )
    §14.9(ii) Connections Between 𝖯 ν ± μ ( ± x ) , 𝖰 ν μ ( ± x ) , 𝖰 ν μ ( x )
    §14.9(iii) Connections Between P ν ± μ ( x ) , P ν 1 ± μ ( x ) , 𝑸 ν ± μ ( x ) , 𝑸 ν 1 μ ( x )
    §14.9(iv) Whipple’s Formula
    10: Bibliography T
  • Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
  • N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.