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11: 36.5 Stokes Sets
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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 consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set.
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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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12: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Special value of the hypergeometric function and connection formulae among asymptotic expansions.
J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
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13: 30.9 Asymptotic Approximations and Expansions
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§30.9(i) Prolate Spheroidal Wave Functions
… ►§30.9(ii) Oblate Spheroidal Wave Functions
… ►§30.9(iii) Other Approximations and Expansions
… ►The asymptotic behavior of and as is given in Erdélyi et al. (1955, p. 151). …14: 26.2 Basic Definitions
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►A lattice path is a directed path on the plane integer lattice .
Unless otherwise specified, it consists of horizontal segments corresponding to the vector and vertical segments corresponding to the vector .
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►A k-dimensional lattice path is a directed path composed of segments that connect vertices in so that each segment increases one coordinate by exactly one unit.
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►See Table 26.2.1 for .
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15: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Pagurova (1963) tabulates and (with different notation) for , to 7D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
16: 36 Integrals with Coalescing Saddles
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17: 8 Incomplete Gamma and Related
Functions
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18: 28 Mathieu Functions and Hill’s Equation
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19: 23 Weierstrass Elliptic and Modular
Functions
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