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11: 10.72 Mathematical Applications
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§10.72(i) Differential Equations with Turning Points
… ►Simple Turning Points
… ►Multiple or Fractional Turning Points
… ►§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point
►In (10.72.1) assume and depend continuously on a real parameter , has a simple zero and a double pole , except for a critical value , where . …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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Normal forms of functions near degenerate critical points, the Weyl groups and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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13: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves.
J. Phys. A 10 (12), pp. 2061–2081.
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A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping.
Stud. Appl. Math. 101 (4), pp. 433–455.
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More than 41% of the zeros of the zeta function are on the critical line.
Acta Arith. 150 (1), pp. 35–64.
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Density profiles in confined critical systems and conformal invariance.
Phys. Rev. Lett. 66 (7), pp. 895–898.
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14: 2.4 Contour Integrals
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§2.4(iv) Saddle Points
… ►§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
… ►§2.4(vi) Other Coalescing Critical Points
… ►For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. …15: 25.17 Physical Applications
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►Analogies exist between the distribution of the zeros of on the critical line and of semiclassical quantum eigenvalues.
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►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)).
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16: 36.11 Leading-Order Asymptotics
17: Bibliography V
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On the zeros of the Riemann zeta function in the critical strip. IV.
Math. Comp. 46 (174), pp. 667–681.
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On the method of saddle points.
Appl. Sci. Research B. 2, pp. 33–45.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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18: 25.18 Methods of Computation
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