Reflected functions and periodicity

Abstract

In the article, some non-regular F(p) field is determined (the non-regular function of the p=x+iy variables). We obtain a main result after consideration of the F(p) values: F(p)=f(p-2A), if p=x+iy, x=A, for all  A.  The F(A+iy) values are equal to the f(-A+iy) values  in the A+iy point as a result of moving f(p) function to the right for all y on the 2A distance. A reflection of the F(p) field in the (A,0) point is the f(-p) regular function for all real A. For some functions, the result of the moving is equal to the result of the reflection, and we obtain the f(z-2A)=f(-z) equality for all the (A,y) points. After the change of order of axes of coordinates, it is possible to prove, that the F(p) values (in the right part of the plane) are equal to the f(p) values (for the regular f(p) function). We obtain, that the f(p) function is periodic.

       The result of the moving is equal to the result of the reflection, and we obtain the f(z-2A)=f(-z) equality for the (A,y) points, if f(A+w)=f(A-w) (we use, that the Re z=-A line passes to the values of the same analytical function as for the reflection so as for the moving). In this situation, the F(p) values (in the right part of the plane) are equal to the f(p) values (for the regular f(p) function).  

      We can use the f(p)=u+iv equality, if F(p)=u-iv (for the regular f(p) functions with the real values on the imaginary axis). We get  u(x-2A,y)=u(x,y)=u(-x,y) for all A=x>0 too, and  the U(x,y) field is equal to the  harmonic u(x,y) function.  By the definition, the U(A,y) values are equal to the u(x-2A,y) values in the A+iy point as a result of  moving of the u(x,y) function to the right for all y on the 2A distance.  The U(x,y) field is the field of the moving function too. It is proved, that the u(x,0)/dx function is equal to 0 on the complex axis, if u(x,y)=u(-x,y). In this situation, the f(p) values are real in the right part of the plane. The continuation of the regular functions is possible for the u(x,y)+iv(x,y) functions. From the point of some new metrics, the length of vectors depends on the direction of the vectors. It is some possible explanation of the results of the article.