Euler sums (first, second, third)
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21—25 of 25 matching pages
21: Bibliography O
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Summing one- and two-dimensional series related to the Euler series.
J. Comput. Appl. Math. 98 (2), pp. 245–271.
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Studies on the Painlevé equations. IV. Third Painlevé equation
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Funkcial. Ekvac. 30 (2-3), pp. 305–332.
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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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Second-order differential equations with fractional transition points.
Trans. Amer. Math. Soc. 226, pp. 227–241.
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Algorithm 22: Riccati-Bessel functions of first and second kind.
Comm. ACM 3 (11), pp. 600–601.
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22: 32.8 Rational Solutions
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(c)
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§32.8(ii) Second Painlevé Equation
… ►The first four are … ►§32.8(iii) Third Painlevé Equation
… ►In the general case assume , so that as in §32.2(ii) we may set and . … ►, , and , with even.
23: 11.2 Definitions
24: Bibliography F
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The third Appell function for one large variable.
J. Approx. Theory 165, pp. 60–69.
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation.
Math. Comp. 19 (89), pp. 97–104.
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Euler sums and contour integral representations.
Experiment. Math. 7 (1), pp. 15–35.
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Numerical evaluation of the elliptic integral of the third kind.
Math. Comp. 19 (91), pp. 494–496.
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25: 10.17 Asymptotic Expansions for Large Argument
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►Corresponding expansions for other ranges of can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4).
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►Then the remainder associated with the sum
does not exceed the first neglected term in absolute value and has the same sign provided that .
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►If these expansions are terminated when , then the remainder term is bounded in absolute value by the first neglected term, provided that .
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►where ; see §9.7(i).
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►where is the incomplete gamma function (§8.2(i)).
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