Entire Functions Of Infinite Order Research Articles

Asymptotic Values of Entire Functions of Infinite Order

We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.

A growth aspect of entire functions of infinite order

A growth aspect of entire functions of infinite order

On the number of linearly independent admissible solutions to linear differential and linear difference equations

AbstractA classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then each solution base of the differential equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least $n-p$ entire functions of infinite order. Here, the transcendental coefficient $A_p$ dominates the growth of the polynomial coefficients $A_{p+1},\ldots ,A_{n-1}$ . By expressing the dominance of $A_p$ in different ways and allowing the coefficients $A_{p+1},\ldots ,A_{n-1}$ to be transcendental, we show that the conclusion of Frei’s theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that $0$ is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich’s theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex q-difference equations.

Wandering domains for entire functions of finite order in the Eremenko–Lyubich class

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.

On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets

The regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generalized bi-axially symmetric potentials (GBSP's). McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}(F;D),(p\ge 2,D$ is parabolic-convex set) tends to zero depends on the order of $GBSP$ F and obtained a formula for finite order. If $GBSP$ F is an entire function of infinite order then above formula fails to give satisfactory information about the rate of decrease of $E^{\frac{p}{2n}}_{2n}(F;D)$. The purpose of the present work is to refine above result by using the concept of index-q. Also, the formula corresponding to $q$-order does not always hold for lower $q$-order. Therefore we have proved a result for lower $q$-order also, which have not been studied so far.

On the magnitudes of deviations of entire functions of infinite order from rational functions

The magnitudes of deviations b(a, f) of entire functions of infinite order from rational functions are studied.

Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf (k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.

ON CONJECTURE OF R. BR\"{U}CK CONCERNING THE ENTIRE FUNCTION SHARING ONE VALUE CM WITH ITS DERIVATIVE

In this paper, we investigate the conjecture of R. Br\{u}ck, and prove that the conjecture of R. Br\{u}ck holds for entire functions of infinite order and hyper order less than $\frac{1}{2}.$

Real entire functions of infinite order and a conjecture of Wiman

We prove that if f is a real entire function of infinite order, then ff’’ has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff’’ has only real zeros, then f is in the Laguerre-Polya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.

Estimates of the growth and decrease on curves of an entire function of infinite order

Relations between the growth of an entire Dirichlet series with Fejér lacunae and its decrease on curves going to infinity are studied. Sharp estimates are found in the most general case.

BEHAVIOR OF THE LOGARITHM OF THE MODULUS OF THE SUM OF A DIRICHLET SERIES CONVERGING IN A HALF-PLANE

A stronger version of the well-known theorem of Hayman concerning the minimum modulus of an entire function of infinite order defined by a lacunary power series is proved for series converging only in the unit disc or in a half-plane.

The Rate of Convergence of Fourier Coefficients for Entire Functions of Infinite Order with Application to the Weideman-Cloot Sinh-Mapping for Pseudospectral Computations on an Infinite Interval

We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z2) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z).

On the explicit determination of certain solutions of periodic differential equations

We consider the class of equations of the form (∗)w′′ + A(z)w = 0, where A(z) is a nonconstant entire periodic function (where, for convenience, we take the period to be 27φi), and where we assume that A(z) is a rational function of ez . The solutions of such equations are entire functions of infinite order of growth, but such equations can possess one or two linearly independent solutions each of whose zero-sequences has a finite exponent of convergence. In this paper, we develop a method which involves using only the simple technique of undetermined coefficients, which can test any equation (∗) for the existence of a solution whose zero-sequence has a finite exponent of convergence. The method will also produce explicitly any such solution.

Zeros of entire functions of infinite order

Zeros of entire functions of infinite order

On the oscillation of solutions of certain linear differential equations in the complex domain

Our starting point is the differential equationwhere A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],for all , b∈C\{0}, at least outside a set of r of finite measure.

The Reciprocal of an Entire Function of Infinite Order ond the distribution of the Zeros of its Second Derivative

The Reciprocal of an Entire Function of Infinite Order ond the distribution of the Zeros of its Second Derivative

The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative

Let f f be a real entire function of infinite order whose zeros together with those of f ′ f’ are all real. It is proved that ( 1 / f ) (1/f) has an infinity of nonreal zeros. The location of the zeros of f f and ( 1 / f ) (1/f) is also investigated. The result complements a finite order result of Hellerstein and Williamson.

On entire functions of infinite order with radially distributed zeros

On entire functions of infinite order with radially distributed zeros

Set of asymptotic values of a meromorphic function of finite order

It is well known that an entire function of finite order has an at most countable number of asymptotic values. Gross [I] constructed an example of an entire function of infinite order whose set of asymptotic values coincides with the extended complex plane. Dreisin and Weizmann [2] raised the question of whether there are any restrictions from above on the size of the set of asymptotic values of a meromorphic function of finite order. In this direction only the result of Valiron [3] is known; he constructed a meromorphic functlon of finite order with a set of asymptotic values having the cardinality of the continuum 9 In this paper we construct a meromorphic function of finite order whose set of asymptotic values coincides with the extended complex plane, thereby giving a negative answer to the question of Dreisin and Weizmann.

Derivatives of entire functions and a question of Pólya. II

It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with f ′ f’ , only real zeros, then f f has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which f , f ′ f,f’ , and f f have only real zeros are those in the Laguerre-Pólya class, i.e. \[ f ( z ) = z m exp ⁡ { − a z 2 + b z + c } ∏ n ( 1 − z z n ) e z / z n , f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}, \] a ⩾ 0 , b , c a \geqslant 0,b,c and the z n {z_n} real, and Σ z n − 2 > ∞ \Sigma z_n^{ - 2} > \infty . This gives a strong affirmative version of an old conjecture of Pólya.