with%20complex%20parameter
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1: 25.11 Hurwitz Zeta Function
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has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue .
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25.11.13
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§25.11(xii) -Asymptotic Behavior
… ►Similarly, as in the sector , …2: 15.10 Hypergeometric Differential Equation
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15.10.3
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15.10.5
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15.10.7
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►(c) If the parameter
in the differential equation equals , then fundamental solutions in the neighborhood of are given by times those in (a) and (b), with and replaced throughout by and , respectively.
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►The connection formulas for the principal branches of Kummer’s solutions are:
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3: 36.2 Catastrophes and Canonical Integrals
4: 28.16 Asymptotic Expansions for Large
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28.16.1
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5: 20.7 Identities
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►See Lawden (1989, pp. 19–20).
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§20.7(viii) Transformations of Lattice Parameter
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20.7.28
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20.7.29
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20.7.34
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6: 12.11 Zeros
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12.11.9
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7: 30.9 Asymptotic Approximations and Expansions
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►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986).
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►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995).
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►The behavior of for complex
and large is investigated in Hunter and Guerrieri (1982).
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30.9.1
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8: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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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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Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions.
Comput. Phys. Comm. 178 (7), pp. 535–551.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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9: 11.6 Asymptotic Expansions
10: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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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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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I.
Technical report
Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
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