The Properties Of Solutions Of Linear Differential Equations In Complex Domain

In this paper, we use Nevanlinna' value distribution theory and Wiman-Valiron theory to getinsight into the properties of solutions of linear differential equations in complex domain.In Chapterâ… , we introduce the history and methods of complex linear differential equations,and emphasize the topics that we will relate in chapterâ…¡-â…¤, in which we apply the Nevanlinna'value distribution theory to discussing the properties of solutions of linear differential equations incomplex domain.In Chapterâ…¡, we mainly investigate the zeros distribution of solutions of second order lineardifferential equation f″+A(z)f=0,in an angular domainΩ(α,β)={z|α≤argz≤β,|z|＞0}, where A(z) is an entire function. Arelation between a cluster ray of zeros and Borel direction of E with hyper order+∞is established,where E=f₁f₂, f₁ and f₂ are two linearly independent solutions of f″+A(z)f=0.In Chapterâ…¢, we discuss the zeros distribution of solutions of equation f⁽ⁿ⁾+A_n-2(z)f⁽(n-2)+…+A₁(z)f′+A₀(z)f=0(n≥2),where A₀(z), A₁(z),…, A_n-2(z) are entire functions with orderσ(A_j)=+∞and hyper orderσ₂(A_j)=0(j=0,1,2,…,n-2)in an angular domainΩ(α,β), and obtain a relation between acluster ray of zeros and Borel direction of E with hyper order +∞, where E=f₁f₂…f_n, andf₁, f₂,…, f_n be n linearly independent, solutions of the differential equation f⁽ⁿ⁾+A_n-2(z)f^(n-2)+…+A₁(z)f′+A₀(z)f=0(n≥2).In Chapterâ…£, we mainly obtain representations of subnormal solutions of second order lineardifferential and higher order linear differential equations with periodic coefficients which are poly-nomials in e^z and e^-z, and resolve completely the problem raised by G.G. Gundersen and E.M.Steinbat in 1994.In Chapterâ…¤, we give the properties of e-type order and e-type exponent of convergence ofzeros of an entire function, the properties of Nevanlinna characteristic function in |z|＞R₀, and theperturbation results that have been done by Chiang and Gao, and obtain the perturbation resultsof f″+Π(z)A(z)f=0 and f″+{Π(z)A(z)+exp(P(e^z))}f=0, where A(z) andΠ(z) are periodicentire functions with period 2Ï€i, P(e^z) is a polynomial in e^z of degree n, andρ_e(Π)＜ρ_e(A).