About the Project
NIST

zeros of analytic functions

AdvancedHelp

(0.006 seconds)

11—20 of 40 matching pages

11: 22.2 Definitions
§22.2 Definitions
Each is meromorphic in z for fixed k , with simple poles and simple zeros, and each is meromorphic in k for fixed z . … … The Jacobian functions are related in the following way. … In terms of Neville’s theta functions20.1) …
12: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). …
13: Bibliography W
  • P. L. Walker (2003) The analyticity of Jacobian functions with respect to the parameter k . Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
  • P. L. Walker (2007) The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.
  • P. L. Walker (2009) The distribution of the zeros of Jacobian elliptic functions with respect to the parameter k . Comput. Methods Funct. Theory 9 (2), pp. 579–591.
  • H. S. Wall (1948) Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.
  • J. Wimp (1965) On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.
  • 14: 28.7 Analytic Continuation of Eigenvalues
    §28.7 Analytic Continuation of Eigenvalues
    As functions of q , a n ( q ) and b n ( q ) can be continued analytically in the complex q -plane. …In consequence, the functions can be defined uniquely by introducing suitable cuts in the q -plane. … All the a 2 n ( q ) , n = 0 , 1 , 2 , , can be regarded as belonging to a complete analytic function (in the large). Therefore w I ( 1 2 π ; a , q ) is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). …
    15: 16.4 Argument Unity
    The function F 2 3 ( a , b , c ; d , e ; 1 ) is analytic in the parameters a , b , c , d , e when its series expansion converges and the bottom parameters are not negative integers or zero. …
    16: 2.8 Differential Equations with a Parameter
    In Case I there are no transition points in D and g ( z ) is analytic. In Case II f ( z ) has a simple zero at z 0 and g ( z ) is analytic at z 0 . … The transformation is now specialized in such a way that: (a) ξ and z are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting ψ ( ξ ) (or part of ψ ( ξ ) ) has solutions that are functions of a single variable. … in which ξ ranges over a bounded or unbounded interval or domain Δ , and ψ ( ξ ) is C or analytic on Δ . … For the former f ( z ) has a zero of multiplicity λ = 2 , 3 , 4 , and g ( z ) is analytic. …
    17: Bibliography
  • L. V. Ahlfors (1966) Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.
  • S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
  • M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
  • G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992b) Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.
  • T. M. Apostol (1985b) Note on the trivial zeros of Dirichlet L -functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
  • 18: 8.2 Definitions and Basic Properties
    §8.2 Definitions and Basic Properties
    Normalized functions are: …
    §8.2(ii) Analytic Continuation
    (8.2.9) also holds when a is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …
    §8.2(iii) Differential Equations
    19: 2.7 Differential Equations
    is one at which the coefficients f ( z ) and g ( z ) are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … Hence unless the series (2.7.8) terminate (in which case the corresponding Λ j is zero) they diverge. … Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution w j ( z ) can be continued analytically into any other sector. … In a finite or infinite interval ( a 1 , a 2 ) let f ( x ) be real, positive, and twice-continuously differentiable, and g ( x ) be continuous. …
    20: 33.2 Definitions and Basic Properties
    §33.2(i) Coulomb Wave Equation
    §33.2(ii) Regular Solution F ( η , ρ )
    F ( η , ρ ) is a real and analytic function of ρ on the open interval 0 < ρ < , and also an analytic function of η when - < η < . … As in the case of F ( η , ρ ) , the solutions H ± ( η , ρ ) and G ( η , ρ ) are analytic functions of ρ when 0 < ρ < . Also, e i σ ( η ) H ± ( η , ρ ) are analytic functions of η when - < η < . …