Рассматриваются комплексные аналитические функции в разных системах координат. Доказаны факты, приводящие к периодичности аналитических функций как следствие совпадения данных функций с некоторыми комплексными функциями, совпадающими с симметричным отражением относительно одной из осей системы координат (полями сдвигов). Доказаны теоремы о возникновении второго уравнения одного многообразия решений уравнения 𝑧 = 𝑓(𝑝) (графика) с точки зрения рассмотрения новых систем координат на комплексной плоскости. Доказательство этих теорем опирается только на известные факты математического анализа.

We consider the analytical 𝑧 = 𝑓(𝑝) functions in different coordinate systems. The centers of coordinates of the new systems are located in the (𝐴, 0) points, with the r variable. With point of view of the new coordinate system we obtain some new 𝑦 = 𝑓(𝑥 + 𝐴) equations for the same 𝑦 = 𝑓(𝑥) graph for the only function, (the new equation of the same (𝑧, 𝑓(𝑝)) set of points for the 𝑝 = 𝑥, 𝑧 = 𝑦 complex variables). To prove the fact we can consider the 𝑦 = 𝑓(𝑝) equation for the new 𝑝 = 𝑟 + 𝐴 variable with 𝑟 = 𝑥; by definition of the 𝑝 and 𝑟 variables 𝑝 = 𝑟 + 𝐴, (the 𝑝 variable we consider in the primary coordinate system, the 𝑟 variable we consider in the new coordinate system with the (𝐴, 0) center of coordinate); for all the 𝑟, 𝑥 variables 𝑟 = 𝑥 by definition of the radius-vectors in both coordinate systems, (𝑟 is other designation of 𝑥). The consideration of the equations results in periodicity of the 𝑓(𝑝) function. The same result we obtain for the complex 𝐹(𝑝) field, where 𝐹(𝑥+𝑖𝑦) = 𝑓(−𝑥+𝑖𝑦) by definition for the 𝑝 = 𝑥 + 𝑖𝑦 complex variable. In the new coordinate system the i constant is located on the OX axis instead of the 𝑂𝑌 axis. In the system of coordinates the new equation of the (𝑧, 𝑓(𝑝)) set of points (graph) is the same as in the initial system of coordinates. It is proved, that the 𝑓(𝑖𝑥 + 𝑦) field in relation to the 𝑥 = 𝑦 diagonal is equal to the 𝑓(−𝑖(−𝑥 + 𝑖𝑦)) field in relation to the 𝑖𝑂𝑋 axis.