scholarly journals Propagation sets of holomorphic curves

2020 ◽  
pp. 2050126
Author(s):  
Jianhua Zheng ◽  
Qiming Yan
Keyword(s):  
Finite Order ◽  
Complex Plane ◽  
Linear Relation ◽  
Infinite Order ◽  
Holomorphic Curves

We consider a problem of whether a property of holomorphic curves on a subset [Formula: see text] of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is the uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, [Formula: see text] is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in [Formula: see text] can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks. (In fact, the method applies to the case of finite order.) For a general case, the union of a sequence of annuli will be a propagation set. The classic five-value theorem and four-value theorem of Nevanlinna are established in such propagation sets.

scholarly journals The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Wen Ju Tang ◽  
Jian Chen ◽  
Hong Yan Xu
Keyword(s):  
Finite Order ◽  
Complex Plane ◽  
Infinite Order ◽  
Entire Functions ◽  
Growth Order ◽  
Stieltjes Transform ◽  
Proximate Order ◽  
Growth Of Entire Functions ◽  
Stieltjes Transforms

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

1986 ◽  
Vol 102 (1-2) ◽  
pp. 9-17 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Infinite Order ◽  
Independent Interest ◽  
Partial Result ◽  
Transcendental Entire Function ◽  
Degree Polynomial ◽  
Odd Degree ◽  
Polynomial Of Odd Degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

scholarly journals Lebesgue measure of escaping sets of entire functions

2018 ◽  
Vol 40 (1) ◽  
pp. 89-116 ◽  
Author(s):  
WEIWEI CUI
Keyword(s):  
Entire Function ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Finite Order ◽  
Infinite Order ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Positive Lebesgue Measure

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

scholarly journals Further Results about a Special Fermat-Type Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Hongwei Ma ◽  
Jianming Qi ◽  
Zhenjie Zhang
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Complex Plane ◽  
Meromorphic Solution ◽  
Unique Range Set ◽  
The Difference

In this paper, we prove the difference equation Fz3+ΔcFz+c3=1 does not have meromorphic solution of finite order over the complex plane C. We also discuss an application to the unique range set problem.

A note on integral functions

1932 ◽  
Vol 28 (3) ◽  
pp. 262-265 ◽  
Author(s):  
R. E. A. C. Paley
Keyword(s):  
Finite Order ◽  
Infinite Order ◽  
Integral Function ◽  
Zero Order ◽  
Image Position

1. Let f(z) denote an integral function of finite order ρ. We writeIt has been shown thatwhere hρ is a constant which depends only on ρ. We are naturally led to enquire whether some equation of the form (1.1) may be true with lim sup replaced by lim inf. In this note we show that the reverse is true. We construct an integral function of zero order for whichThe proof may easily be modified to construct a function of any finite order or of infinite order for which (1.2) is satisfied.

scholarly journals Cohomology of the variational complex in the class of exterior forms of finite jet order

2002 ◽  
Vol 30 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Gennadi Sardanashvily
Keyword(s):  
Inverse Problem ◽  
Finite Order ◽  
Calculus Of Variations ◽  
Fiber Bundle ◽  
Infinite Order ◽  
Fiber Bundles ◽  
Jet Space ◽  
Variational Complex ◽  
Exterior Forms

We obtain the cohomology of the variational complex on the infinite-order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. In particular, this provides a solution of the global inverse problem of the calculus of variations of finite order on fiber bundles.

Testing for Causation Using Infinite Order Vector Autoregressive Processes

1996 ◽  
Vol 12 (1) ◽  
pp. 61-87 ◽  
Author(s):  
Helmut Lütkepohl ◽  
D.S. POSKITT
Keyword(s):  
Granger Causality ◽  
Finite Order ◽  
Infinite Order ◽  
Empirical Studies ◽  
Asymptotic Properties ◽  
General Assumption ◽  
Small Sample ◽  
Order Process ◽  
Vector Autoregressive ◽  
Vector Autoregressive Processes

Tests for Granger-causality have been performed in numerous empirical studies. These tests are usually based on finite order vector autoregressive (VAR) processes, and the assumption is made that the model fitted to the available data corresponds to the true data generating mechanism. In the present study, the more general assumption is made that a finite order VAR model is fitted to a potentially infinite order process. The order is assumed to increase with the sample size. Asymptotic properties of tests for Granger-causality as well as other types of causality concepts are derived. Some limited small sample results are obtained using simulation methods.

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