scholarly journals On the complex oscillation theory of f(k) + Af = F

1993 ◽  
Vol 36 (3) ◽  
pp. 447-461 ◽  
Author(s):  
Chen Zong-Xuan
Keyword(s):  
Entire Functions ◽  
Oscillation Theory ◽  
Order Of Growth ◽  
Exponent Of Convergence ◽  
Complex Oscillation ◽  
Zero Sequence ◽  
Image Position ◽  
Growth Of Solutions

In this paper, we investigate the complex oscillation theory ofwhere A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.

scholarly journals A further result on the complex oscillation theory of periodic second order linear differential equations

1990 ◽  
Vol 33 (1) ◽  
pp. 143-158 ◽  
Author(s):  
Shian Gao
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Oscillation Theory ◽  
Transcendental Entire Function ◽  
Arbitrary Positive Integer ◽  
Order Of Growth ◽  
Exponent Of Convergence ◽  
Complex Oscillation ◽  
Zero Sequence ◽  
Linear Differential

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

scholarly journals Scalar and matrix complex nonoscillation criteria

1973 ◽  
Vol 18 (3) ◽  
pp. 173-189
Author(s):  
H. C. Howard
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Recent Work ◽  
Complex Variable ◽  
Oscillation Theory ◽  
Recent Book ◽  
Complex Oscillation ◽  
Special Cases ◽  
Image Position ◽  
Mathieu Equations

In his recent book on ordinary differential equations Hille (3) devotes a chapter to complex oscillation theory. Drawing upon his own work in this area and the work of Nehari, Schwarz, Taam, and others, he gives a variety of oscil-lation and nonoscillation theorems for solutions of the differential equationwhere z is a complex variable and p is regular in some appropriate domain. There are a number of results for (1.1) with an arbitrary coefficient o and some discussions for special cases of classical interest, such as the Bessel and Mathieu equations. There is a bibliography at the end of the chapter. For other recent work in this area attention is directed to papers by Herold (1, 2) Kim (4, 5) and Lavie (6) where other references are given.

On complex oscillation, function-theoretic quantization of non-homogeneous periodic ordinary differential equations and special functions

2012 ◽  
Vol 142 (3) ◽  
pp. 449-477
Author(s):  
Yik-Man Chiang ◽  
Kit-Wing yu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Special Functions ◽  
Sufficient Conditions ◽  
Main Idea ◽  
Oscillation Theory ◽  
Exponent Of Convergence ◽  
Oscillatory Solutions ◽  
Complex Oscillation ◽  
Necessary And Sufficient

New necessary and sufficient conditions are given for the quantization of a class of periodic second-order non-homogeneous ordinary differential equations in the complex plane. The problem is studied from the viewpoint of complex oscillation theory first developed in works by Bank and Laine and Gundersen and Steinbart. We show that, when a solution is complex non-oscillatory (finite exponent of convergence of zeros), then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch, and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are polynomial/rational-type functions, which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a recent paper by Chiang and Yu. In addition to the asymptotics of special functions, the other main idea that we apply in our proof is a classical result by Wright that gives precise asymptotic locations of large zeros of a functional equation.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

Representation Formulas for Integrable and Entire Functions of Exponential Type I

1988 ◽  
Vol 40 (04) ◽  
pp. 1010-1024 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Conjugate Function ◽  
Interpolation Formula ◽  
Type I ◽  
Additional Hypothesis ◽  
Period 2 ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

Let Bτ denote the class of entire functions of exponential type τ (&gt;0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143] 1.1 where t, γ are real numbers and is the so called conjugate function of f. Let us put 1.2 The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε &gt; 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

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