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11: 13.27 Mathematical Applications
where α , β , γ , δ are real numbers, and γ > 0 . Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. … For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
12: 27.5 Inversion Formulas
The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). The multiplicative functions are a subgroup of this group. … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: … For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964). …
13: Bibliography M
  • I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.
  • Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
  • W. Magnus (1941) Zur Theorie des zylindrisch-parabolischen Spiegels. Z. Physik 118, pp. 343–356 (German).
  • N. W. McLachlan (1934) Loud Speakers: Theory, Performance, Testing and Design. Oxford University Press, New York.
  • L. J. Mordell (1958) On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.
  • 14: 27.14 Unrestricted Partitions
    Euler’s pentagonal number theorem states that …where the exponents 1 , 2 , 5 , 7 , 12 , 15 , are the pentagonal numbers, defined by … For example, the Ramanujan identity …
    §27.14(vi) Ramanujan’s Tau Function
    15: Bibliography B
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • D. Bleichenbacher (1996) Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.
  • J. M. Borwein and P. B. Borwein (1987) Pi and the AGM, A Study in Analytic Number Theory and Computational Complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York.
  • D. Bressoud and S. Wagon (2000) A Course in Computational Number Theory. Key College Publishing, Emeryville, CA.
  • 16: Bibliography V
  • H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ (1988) Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.
  • N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
  • M. N. Vrahatis, O. Ragos, T. Skiniotis, F. A. Zafiropoulos, and T. N. Grapsa (1997b) The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.
  • 17: Bibliography K
  • E. Kanzieper (2002) Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).
  • T. Kim and H. S. Kim (1999) Remark on p -adic q -Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.
  • A. Kneser (1927) Neue Untersuchungen einer Reihe aus der Theorie der elliptischen Funktionen. Journal für die Reine und Angenwandte Mathematik 158, pp. 209–218 (German).
  • K. Knopp (1948) Theory and Application of Infinite Series. 2nd edition, Hafner, New York.
  • M. Koecher (1954) Zur Theorie der Modulformen n -ten Grades. I. Math. Z. 59, pp. 399–416 (German).
  • 18: Bibliography S
  • M. R. Schroeder (2006) Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. 4th edition, Springer-Verlag, Berlin.
  • D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
  • D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.
  • D. Sornette (1998) Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.
  • A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 19: Mathematical Introduction
    The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … The first three chapters of the NIST Handbook and DLMF are methodology chapters that provide detailed coverage of, and references for, mathematical topics that are especially important in the theory, computation, and application of special functions. …
    ( a , b ] or [ a , b )

    half-closed intervals.

    set of all positive integers.

    n

    set of all integer multiples of n .

    For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …