Existence of an entire transcendental function of boundedl-index

1995 ◽  
Vol 57 (1) ◽  
pp. 88-90 ◽  
Author(s):  
A. A. Gol'dberg ◽  
M. N. Sheremeta
Keyword(s):  
Transcendental Function ◽  
Entire Transcendental Function

scholarly journals Dynamics of exp (z)

1984 ◽  
Vol 4 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Robert L. Devaney ◽  
Michal Krych
Keyword(s):  
Complex Plane ◽  
Symbolic Dynamics ◽  
Final Section ◽  
Julia Set ◽  
Dynamical Behaviour ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Orbit Structure ◽  
The Family ◽  
Entire Complex

AbstractWe describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.

scholarly journals Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 43
Author(s):  
Myroslav Sheremeta
Keyword(s):  
Asymptotic Behavior ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Relative Growth ◽  
Type Integral ◽  
Stieltjes Type

For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr→+∞α(Mf−1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=∫0∞a(x)f(rx)dF(x). Unresolved problems are formulated.

scholarly journals An arithmetic remark on entire periodic functions

1971 ◽  
Vol 5 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Transcendental Function ◽  
Periodic Functions ◽  
Entire Transcendental Function ◽  
Image Position ◽  
Rational Integral

For every positive number w, there exists an odd entire transcendental function.with rational integral coefficients ah such that f〈z+w〉 = f〈z〉.

Non-Real Periodic Points of Entire Functions

1997 ◽  
Vol 40 (3) ◽  
pp. 271-275 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Complex Plane ◽  
Entire Functions ◽  
Periodic Points ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Straight Line

AbstractIt is shown that if f is an entire transcendental function, l a straight line in the complex plane, and n ≥ 2, then f has infinitely many repelling periodic points of period n that do not lie on l.

Invariant domains and singularities

1995 ◽  
Vol 117 (3) ◽  
pp. 525-532 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Jordan Curve ◽  
Euclidean Distance ◽  
The Other ◽  
Transcendental Function ◽  
Fatou Set ◽  
Entire Transcendental Function ◽  
Asymptotic Values ◽  
Other Hand ◽  
Invariant Component ◽  
Invariant Domains

AbstractLet U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.

scholarly journals Singular values and bounded Siegel disks

2017 ◽  
Vol 165 (2) ◽  
pp. 249-265
Author(s):  
ANNA MIRIAM BENINI ◽  
NÚRIA FAGELLA
Keyword(s):  
Critical Point ◽  
Finite Order ◽  
Rotation Number ◽  
General Theorem ◽  
Singular Values ◽  
Transcendental Function ◽  
Siegel Disk ◽  
Entire Transcendental Function

AbstractLet f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

scholarly journals Some entire functions with fixpoints of every order

1960 ◽  
Vol 1 (2) ◽  
pp. 203-209 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Complex Variable ◽  
Entire Functions ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Transcendental Functions ◽  
Image Position

In this paper f(z) will always stand for an entire transcendental function of the complex variable z. For p= 1, 2, … the natural iterate fD(z) of f(z) is defined by These natural iterates are themselves entire transcendental functions; they have been studied by various writers, notably Fatou [3]. References to many papers on iterated will be found in [1].

scholarly journals On the growth of series in system of functions and Laplace-Stieltjes integrals

2021 ◽  
Vol 55 (2) ◽  
pp. 124-131
Author(s):  
M.M. Sheremeta
Keyword(s):  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Type Integral ◽  
System F ◽  
The Relationship ◽  
Stieltjes Type ◽  
Generalized Order

For a regularly convergent in ${\Bbb C}$ series $A(z)=\sum\nolimits_{n=1}^{\infty}a_nf(\lambda_nz)$ in the system ${f(\lambda_nz)}$, where$f(z)=\sum\nolimits_{k=0}^{\infty}f_kz^k$ is an entire transcendental function and $(\lambda_n)$is a sequence of positive numbers increasing to $+\infty$, it isinvestigated the relationship between the growth of functions $A$ and $f$ in terms of a generalized order. It is proved that if$a_n\ge 0$ for all $n\ge n_0$, $\ln \lambda_n=o\big(\beta^{-1}\big(c\alpha(\frac{1}{\ln \lambda_n}\ln \frac{1}{a_n})\big)\big)$ for each $c\in (0, +\infty)$ and $\ln n=O(\Gamma_f(\lambda_n))$ as $n\to\infty$ then $\displaystyle\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_A(r))}{\beta(\ln r)}=\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_f(r))}{\beta(\ln r)},$ where $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, $\Gamma_f(r):=\frac{d\ln M_f(r)}{d\ln r}$ and positive continuous on $(x_0, +\infty)$ functions $\alpha$and $\beta$ are such that $\beta((1+o(1))x)=(1+o(1))\beta(x)$, $\alpha(c x)=(1+o(1))\alpha(x)$ and$\frac{d\beta^{-1}(c\alpha(x))}{d\ln x}=O(1)$ as $x\to+\infty$ for each $c\in(0, +\infty)$.\A similar result is obtained for the Laplace-Stieltjes type integral $I(r) = \int\limits_{0}^{\infty}a(x)f(rx) dF(x)$.

Conversion of Transcendental Function into Imprecise Function

2018 ◽  
Vol 9 (6) ◽  
pp. 501-512
Author(s):  
Kangujam Priyokumar Singh ◽  
Sahalad Borgoyary
Keyword(s):  
Transcendental Function
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