# connection formulas across transition points

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##### 1: 36.4 Bifurcation Sets

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###### §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

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###### §2.8(ii) Case I: No Transition Points

… ►For error bounds, more delicate error estimates, extensions to complex $\xi $ 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: 12.16 Mathematical Applications

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►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).
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►In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs.
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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##### 4: 36.5 Stokes Sets

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►where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu $ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re}\mathrm{\Phi}=\mathrm{constant}$) in complex $t$ or $(s,t)$ space.
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►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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►This part of the Stokes set connects two complex saddles.
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►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
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##### 5: Bibliography O

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Fourier Expansions. A Collection of Formulas.
Academic Press, New York-London.
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Connection formulas for second-order differential equations with multiple turning points.
SIAM J. Math. Anal. 8 (1), pp. 127–154.
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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.
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Second-order differential equations with fractional transition points.
Trans. Amer. Math. Soc. 226, pp. 227–241.
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General connection formulae for Liouville-Green approximations in the complex plane.
Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
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##### 6: Bibliography W

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Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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On the central connection problem for the double confluent Heun equation.
Math. Nachr. 195, pp. 267–276.
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On the connection formulas of the fourth Painlevé transcendent.
Anal. Appl. (Singap.) 7 (4), pp. 419–448.
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On the connection formulas of the third Painlevé transcendent.
Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.
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##### 7: 31.18 Methods of Computation

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►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).
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##### 8: 14.21 Definitions and Basic Properties

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${P}_{\nu}^{\pm \mu}\left(z\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(z\right)$ exist for all values of $\nu $, $\mu $, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty}$, which are branch points (or poles) of the functions, in general.
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###### §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: Bibliography T

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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis.
Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
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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.
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Explicit formulas for the Bernoulli and Euler polynomials and numbers.
Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
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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.
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##### 10: 9.16 Physical Applications

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►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood.
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►These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability.
The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)).
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►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation).
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►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
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