Some Relations between Certain Complex Equations and Nonnormalized Meromorphic Functions

The purpose of this investigation is first to reveal some relations between certain complex (differential) equations and nonnormalized meromorphic functions and then to point some of their useful consequences out. 1. Introduction, Notations, Definitions, and Motivation As is known, a differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called a partial differential equation; if only ordinary derivatives are present, the equation is called an ordinary differential equation (ODE). Differential equations play an extremely important and useful role in applied math, engineering, and physics, and more mathematical and numerical machineries have been developed for the solution to differential equations. As we also know, ODE is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real- or complex-valued function, but more generally, it may be vector-valued or matrix-valued; this corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. There are many methods to compute numerical solutions to differential equations and studies on the properties of differential equations involving approximation theory of the solution to a differential equation by the solution to a corresponding differential equation. In this work, as a novel investigation, we want to focus on certain types of (linear or nonlinear) first-order complex differential equation. Especially, for functions and being analytic in certain domains of complex domain, we want to take cognizance of a few types of certain complex equations including several first-order complex differential equations like and to reveal some of their useful implications between certain complex (differential) equations and nonnormalized multivalent functions which are analytic in the punctured unit disk and