isomonodromy problems
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41: 27.11 Asymptotic Formulas: Partial Sums
§27.11 Asymptotic Formulas: Partial Sums
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27.11.2
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►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number such that the error term in (27.11.2) is for all .
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42: Bibliography G
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Quasirandom distributed bases for bound problems.
J. Chem. Phys. 114 (9), pp. 3929–3939.
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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Problem 72-21, Laplace transforms of Airy functions.
SIAM Rev. 15 (4), pp. 796–798.
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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem.
J. Math. Anal. Appl. 359 (1), pp. 352–367.
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Stirling number representation problems.
Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
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43: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids
►Equation (30.13.7) for together with the boundary condition on the ellipsoid given by , poses an eigenvalue problem with as spectral parameter. …44: DLMF Project News
error generating summary45: 20.9 Relations to Other Functions
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►The relations (20.9.1) and (20.9.2) between and (or ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485).
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46: 3.6 Linear Difference Equations
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►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.
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►A new problem arises, however, if, as , the asymptotic behavior of is intermediate to those of two independent solutions and of the corresponding inhomogeneous equation (the complementary functions).
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►For a difference equation of order (),
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47: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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§30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids
►Equation (30.13.7) for , and subject to the boundary condition on the ellipsoid given by , poses an eigenvalue problem with as spectral parameter. …For the Dirichlet boundary-value problem of the region between two ellipsoids, the eigenvalues are determined from …48: 23.21 Physical Applications
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►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form .
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►Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations.
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49: Bibliography B
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Coefficient functions for an inhomogeneous turning-point problem.
Mathematika 38 (2), pp. 217–238.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Numerical Methods for Least Squares Problems.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model.
Ann. of Math. (2) 150 (1), pp. 185–266.
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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation.
J. Phys. A 30 (2), pp. 559–571.
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50: Bibliography L
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The two-point connection problem for differential equations of the Heun class.
Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).
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The central two-point connection problem for the Heun class of ODEs.
J. Phys. A 31 (18), pp. 4249–4261.
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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics.
J. Math. Phys. 27 (5), pp. 1238–1265.
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Problems of Mathematical Physics.
Revised, enlarged and corrected English edition; translated
and edited by Richard A. Silverman. With a supplement by
Edward L. Reiss, Prentice-Hall Inc., Englewood Cliffs, N.J..
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On incomplete Airy functions and their application to diffraction problems.
Radio Sci. 4 (10), pp. 959–969.
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