On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

2015 ◽  
Vol 67 (1) ◽  
pp. 159-165
Author(s):  
A. A. Mokhon’ko ◽  
A. Z. Mokhon’ko
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Branching Point ◽  
Linear Differential

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.

Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1

1985 ◽  
Vol 100 (3-4) ◽  
pp. 301-326 ◽  
Author(s):  
M. H. Lantsman
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Algebraic Equations ◽  
Order Of Growth ◽  
Constant Coefficients ◽  
Independent Variable ◽  
Linear Differential ◽  
Image Position ◽  
Asymptotic Representations

SynopsisWe consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equationcan be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

scholarly journals Laplace Contour Integrals and Linear Differential Equations

2021 ◽  
Author(s):  
Norbert Steinmetz
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Distribution Of Zeros ◽  
Polynomial Solutions ◽  
Contour Integrals ◽  
Nevanlinna Functions ◽  
Linear Differential

AbstractThe purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ L [ w ] ( z ) : = w ( n ) + ∑ j = 0 n - 1 ( a j + b j z ) w ( j ) = 0 . This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.

Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 2

1985 ◽  
Vol 101 (1-2) ◽  
pp. 77-98 ◽  
Author(s):  
M. H. Lantsman
Keyword(s):  
Differential Equations ◽  
Fundamental System ◽  
Partial Solution ◽  
Linear Differential Equations ◽  
Logarithmic Function ◽  
Order Of Growth ◽  
Logarithmic Coefficients ◽  
Linear Differential ◽  
Logarithmic Functions ◽  
Nonhomogeneous Equation

SynopsisWe consider linear differential equations of the form F(x, z)≡ xn + a1(z)x(n-1)+…+an(z)x = 0 with power-logarithmic coefficients or coefficients which are asymptotically similar to power-logarithmic functions in a central sector S of a complex plane for z →∞, z∈S. The main result of this paper is that in a sufficiently small central sector SE⊂S there is a fundamental system of solutions {xi(z) = exp [∫γi(z)dz)} where each function γi(z) is equivalent to a power-logarithmic function or has an estimate of the form O(z−∞). Furthermore, a precise estimate is obtained for a partial solution of a nonhomogeneous equation F(x, z) = α(z), where the function α(z) grows like a power.

scholarly journals On [p, q]-order of growth of solutions of linear differential equations in the unit disc

2021 ◽  
Vol 6 (11) ◽  
pp. 12878-12893
Author(s):  
Hongyan Qin ◽  
◽  
Jianren Long ◽  
Mingjin Li
Keyword(s):  
Differential Equations ◽  
Analytic Functions ◽  
Unit Disc ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Linear Differential ◽  
Growth Of Solutions

<abstract><p>The $ [p, q] $-order of growth of solutions of the following linear differential equations $ (**) $ is investigated,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{'}+A_{0}(z)f = 0, (**) $\end{document} </tex-math></disp-formula></p> <p>where $ A_{i}(z) $ are analytic functions in the unit disc, $ i = 0, 1, ..., k-1 $. Some estimations of $ [p, q] $-order of growth of solutions of the equation $ (\ast*) $ are obtained when $ A_{j}(z) $ dominate the others coefficients near a point on the boundary of the unit disc, which is generalization of previous results from S. Hamouda.</p></abstract>

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