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11: 22.4 Periods, Poles, and Zeros
The other poles are at congruent points, which is the set of points obtained by making translations by 2 m K + 2 n i K , where m , n . … Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form 2 m K + 2 n i K , where m , n . … The set of points z = m K + n i K , m , n , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by m K + n i K , where again m , n . …
§22.4(iii) Translation by Half or Quarter Periods
12: Bibliography V
  • A. N. Varčenko (1976) Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).
  • N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
  • N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
  • N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
  • A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).
  • 13: Bibliography G
  • I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.
  • S. G. Gindikin (1964) Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).
  • V. V. Golubev (1960) Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..
  • V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).
  • A. Guthmann (1991) Asymptotische Entwicklungen für unvollständige Gammafunktionen. Forum Math. 3 (2), pp. 105–141 (German).
  • 14: Bibliography P
  • V. I. Pagurova (1963) Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).
  • V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
  • B. V. Pal tsev (1999) On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).
  • G. Pólya and R. C. Read (1987) Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag, New York.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986a) Integrals and Series: Elementary Functions, Vol. 1. Gordon & Breach Science Publishers, New York.
  • 15: 20.2 Definitions and Periodic Properties
    §20.2(iii) Translation of the Argument by Half-Periods
    16: 5.19 Mathematical Applications
    By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of f ( z ) for large | z | , or small | z | , can be obtained complete with an integral representation of the error term. …
    17: Bibliography R
  • Ju. M. Rappoport (1979) Tablitsy modifitsirovannykh funktsii Besselya K 1 2 + i β ( x ) . “Nauka”, Moscow (Russian).
  • G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
  • E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
  • 18: Bibliography S
  • J.-P. Serre (1973) A Course in Arithmetic. Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.
  • S. Yu. Slavyanov (1996) Asymptotic Solutions of the One-dimensional Schrödinger Equation. American Mathematical Society, Providence, RI.
  • A. D. Smirnov (1960) Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.
  • V. I. Smirnov (1996) Izbrannye Trudy. Analiticheskaya teoriya obyknovennykh differentsialnykh uravnenii. Izdatel’ stvo Sankt-Peterburgskogo Universiteta, St. Petersburg (Russian).
  • A. Sommerfeld (1928) Atombau und Spektrallinien. Vieweg, Braunschweig.
  • 19: Preface
    Miller was responsible for information architecture, specializing LaTeX for the needs of the project, translation from LaTeX to MathML, and the search interface. …
    20: Bibliography F
  • V. N. Faddeeva and N. M. Terent’ev (1954) Tablicy značeniĭ funkcii w ( z ) = e - z 2 ( 1 + 2 i π 0 z e t 2 d t ) ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
  • V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function w ( z ) = e - z 2 ( 1 + 2 i π - 1 / 2 0 z e t 2 d t ) for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
  • M. V. Fedoryuk (1989) The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).
  • M. V. Fedoryuk (1991) Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).