One opinion about the Riemannian Hypothesis

In the present work is an attempt for a new application of the trigonometric form of complex numbers and functions of complex variables

Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single research paper "Linearizability criteria for systems of two second-order differential equations by complex methods" which has been published in Nonlinear Dynamics. Due to citations of both parts I and II these are not replaced with the above published articl

Дослідження виробничих функцій комплексної змінної

У статті розглядається історичний розвиток комплексних чисел і пропонується дослідження виробничих функцій за допомогою представлення у вигляді функцій комплексних змінних. Для демонстрації деяких прикладів використано пакет програм Mathematica 8. Доведено, що виробничі функції комплексного змінного мають право на існування і значно розширюють інстру- ментальну базу дослідника.The paper considers the historical development of complex numbers; the research of production functions using the representation in the form of functions of complex variables is proposed. To demonstrate some examples the software package Mathematica 8 is used. It is proved that the production functions of complex variables have the right to exist and significantly expand the instrumental basis of the researcher.В статье рассматривается историческое развитие комплексных чисел и предлагается исследование производственных функций с помощью представления в виде функций комплексных переменных. Для демонстрации некоторых примеров используется пакет программ Mathematica 8. Доказано, что производственные функции комплексного переменного имеют право на существование и значительно расширяют инструментальную базу исследователя

Fractal Images Generated by Newton\u27s Method.

We investigate the behavior of Newton\u27s Method for finding roots applied to complex-valued functions of complex variables. This re- quires an analysis of iteration of rational functions. The fractal nature of Newton\u27s Method in the complex plane gives us intricate and beautiful images. By investigating select functions we attempt to generalize a pattern of behavior