expansions%20in%20series%20of
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11: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
… βΊExplicit coefficients in terms of and are given up to in Abramowitz and Stegun (1964, p. 636). βΊFor , and with as in §23.3(i), …Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as . βΊFor …12: 8.17 Incomplete Beta Functions
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βΊHowever, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values.
…where, as in §5.12, denotes the beta function:
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βΊFurther integral representations can be obtained by combining the results given in §8.17(ii) with §15.6.
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βΊThe expansion (8.17.22) converges rapidly for .
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βΊFor sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62).
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13: Bibliography M
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Siegel’s modular forms and Dirichlet series.
Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
…
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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation.
SIAM J. Numer. Anal. 3 (3), pp. 390–409.
…
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A -analog of hypergeometric series well-poised in
and invariant -functions.
Adv. in Math. 58 (1), pp. 1–60.
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A -analog of the Gauss summation theorem for hypergeometric series in
.
Adv. in Math. 72 (1), pp. 59–131.
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14: Bibliography V
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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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Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
…
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Symbolic evaluation of coefficients in Airy-type asymptotic expansions.
J. Math. Anal. Appl. 269 (1), pp. 317–331.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
…
15: Bibliography K
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Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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Asymptotic expansions of certain -series and a formula of Ramanujan for specific values of the Riemann zeta function.
Acta Arith. 107 (3), pp. 269–298.
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Theorie und Anwendung der unendlichen Reihen.
4th edition, Die Grundlehren der mathematischen Wissenschaften, Band 2, Springer-Verlag, Berlin-Heidelberg (German).
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Electromagnetic Fields and Relativistic Particles.
International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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16: Bibliography B
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Singularities in Waves and Rays.
In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.),
Vol. 35, pp. 453–543.
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A short table of the functions , from to
.
Phil. Mag. Series 7 20, pp. 343–347.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Formal Power Series and Algebraic Combinatorics.
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science, Vol. 24, American Mathematical Society, Providence, RI.
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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents.
Methods Appl. Anal. 2 (4), pp. 475–489.
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17: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Elliptic hypergeometric series on root systems.
Adv. Math. 181 (2), pp. 417–447.
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Sources in the development of mathematics.
Cambridge University Press, Cambridge.
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Functional Analysis.
McGraw-Hill Book Co., New York.
βΊ
Principles of Mathematical Analysis.
3rd edition, McGraw-Hill Book Co., New York.
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18: Bibliography C
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On the expansion of a Coulomb potential in spherical harmonics.
Proc. Cambridge Philos. Soc. 46, pp. 626–633.
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Expansions in terms of parabolic cylinder functions.
Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
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The numerical solution of linear differential equations in Chebyshev series.
Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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Asymptotic Expansions.
Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.
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19: Bibliography S
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FGH, a code for the calculation of Coulomb radial wave functions from series expansions.
Comput. Phys. Comm. 146 (2), pp. 250–253.
…
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc. 62, pp. 441–452.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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20: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
.
Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Über die Bestimmung der mittleren Werthe in der Zahlentheorie.
Abhandlungen der Königlich Preussischen Akademie der
Wissenschaften von 1849, pp. 69–83 (German).
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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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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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