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1—10 of 19 matching pages
1: Bibliography G
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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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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs.
SIAM J. Numer. Anal. 41 (3), pp. 827–855.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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2: 32.8 Rational Solutions
§32.8 Rational Solutions
… ►§32.8(ii) Second Painlevé Equation
… ►These solutions have the form … ►These rational solutions have the form … ►3: Bibliography K
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The second Painlevé equation in electric probe theory. Some numerical solutions.
Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface.
Technical Physics 49 (1), pp. 1–7.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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4: Bibliography O
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Hyperasymptotic solutions of second-order linear differential equations. I.
Methods Appl. Anal. 2 (2), pp. 173–197.
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Hyperasymptotic solutions of second-order linear differential equations. II.
Methods Appl. Anal. 2 (2), pp. 198–211.
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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations.
Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.
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Numerical solution of second-order linear difference equations.
J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.
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Bounds for the solutions of second-order linear difference equations.
J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
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5: 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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Rational solutions of Painlevé equations.
Stud. Appl. Math. 61 (1), pp. 31–53.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Perturbation solutions of the ellipsoidal wave equation.
Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
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6: 36.5 Stokes Sets
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36.5.7
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►For , there are two solutions
, provided that .
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►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
The second sheet corresponds to and it intersects the bifurcation set (§36.4) smoothly along the line generated by , .
For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
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7: Bibliography D
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function.
Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
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8: Bibliography V
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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives.
NRL Report No. 6959
Naval Res. Lab. Washingtion, D.C..
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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds.
Math. Comp. 23 (105), pp. 61–69.
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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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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On the rational solutions of the second Painlevé equation.
Differ. Uravn. 1 (1), pp. 79–81 (Russian).
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9: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►On separation of variables into cylindrical coordinates, the Bessel functions , and modified Bessel functions and , all appear.
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►The functions , , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)):
…In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential.
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10: 3.8 Nonlinear Equations
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►Solutions are called roots of the equation, or zeros of .
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►and the solutions are called fixed points of .
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►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013).
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►Consider and .
We have and .
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