We prove the existence of transcendental entire functions f having a property studied by Mahler, namely that f(& Qopf;) subset of & Qopf; and f(-1)(& Qopf;) subset of & Qopf;, and in addition having a ...
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We prove the existence of transcendental entire functions f having a property studied by Mahler, namely that f(& Qopf;) subset of & Qopf; and f(-1)(& Qopf;) subset of & Qopf;, and in addition having a prescribed number of k-periodic algebraic orbits, for all k >= 1. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the ha...
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Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the harmonic sums and their generalizations appear as building blocks, originating for example, from the expansion of generalized hypergeometric functions around integer values of the parameters. In this paper we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system FORM.
The first integrals of the dynamic part of the equations of the axially symmetric solid body moving in a resisting medium in the presence of an additional tracking force were listed completely. In the sense of complex...
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The first integrals of the dynamic part of the equations of the axially symmetric solid body moving in a resisting medium in the presence of an additional tracking force were listed completely. In the sense of complex analysis, the first integrals are the transcendental functions of their variables expressed in terms of a finite combination of the elementary functions.
作者：
Weinzierl, SUniv Parma
Dipartimento Fis Ist Nazl Fis Nucl Grp Collegato Parma I-43100 Parma Italy
Higher transcendental function occur frequently in the calculation of Feynman integrals in quantum field theory. Their expansion in a small parameter is a non-trivial task. We report on a computer program which allows...
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Higher transcendental function occur frequently in the calculation of Feynman integrals in quantum field theory. Their expansion in a small parameter is a non-trivial task. We report on a computer program which allows the systematic expansion of certain classes of functions. The algorithms are based on the Hopf algebra of nested sums. The program is written in C++ and uses the GiNaC library. (C) 2002 Elsevier Science B.V. All rights reserved.
This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at a. This class contains most of the classical families of transcende...
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This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at a. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna's value distribution theory, we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert's metric along with the usual contraction principle. However, these cones allow us to apply a contraction argument stemming from Bowen's initial approach.
Let f : C -> (C) over cap be a transcendental meromorphic function. Suppose that the finite part P(f)boolean AND C of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domai...
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Let f : C -> (C) over cap be a transcendental meromorphic function. Suppose that the finite part P(f)boolean AND C of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded. Then we show that f supports no invariant line fields on its Julia set. We prove this by generalizing two results about rational functions to the transcendental setting: a theorem of Mane (1993) about the branching of iterated preimages of disks, and a theorem of McMullen (1994) regarding the absence of invariant line fields for "measurably transitive" functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Swiatek (2004).
We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orb...
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We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we callshell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, thevirtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type.
We prove a Hadamard-type theorem that associates the generalized order of growth rho(f)* (alpha, beta) of an entire transcendental function f with the coefficients of its expansion in a Faber series. This theorem is a...
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We prove a Hadamard-type theorem that associates the generalized order of growth rho(f)* (alpha, beta) of an entire transcendental function f with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of finite simply connected domain G with boundary. belonging to the Al'per class Lambda*. Using this theorem, we obtain limit equalities that associate rho(f)* (alpha, beta) with a sequence of the best polynomial approximations of f in certain Banach spaces of functions analytic in G.
Series arising from Volterra integral equations of the second kind are summed. The series involve inverse powers of roots of the characteristic equation. It is shown how previous similar series obtained from different...
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Series arising from Volterra integral equations of the second kind are summed. The series involve inverse powers of roots of the characteristic equation. It is shown how previous similar series obtained from differential-difference equations are particular cases of the present development. A number of novel and interesting results are obtained. The techniques are demonstrated through illustrative examples. The application of general techniques to obtain expressions involving the zeros of the Riemann zeta function is investigated. In particular, novel two-parameter expressions are obtained involving the non-trivial zeros of the zeta function.
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on neste...
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Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer. Examples such as expansions of generalized hypergeometric functions or Appell functions are discussed. As a further application, we give the general solution of a two-loop integral, the so-called C-topology, in terms of multiple nested sums. In addition, we discuss some important properties of nested sums, in particular we show that they satisfy a Hopf algebra. (C) 2002 American Institute of Physics.
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