The indefinite expressions in mathematics

Abstract

With help of a new system of coordinates from the point of view of the reverse functions we obtain a two z=f(r+A)=f(p), z=f(r+A)=h(r) equation, which are possible to be considered as the equation of the one M graph on the complex plane, (the set of points M=(z, f(p)): z=f(p) for all p, if p and z are the complex variables), but z=h(r) is the M equation, and z=f(r+A ) is the equation of a other set of point with point of view of the revers functions, (moved to the left on the A>0, for all the values of the complex r=p variable), r is the variable for the (A,0) center of coordinates. In the situation z is the unmoved variable, and p is the other designation of r.

   Similar facts are proved without the use of reverse functions. It is proved, that a two different variables are possible to be considered as one argument in the primary system of coordinates on the complex plane for the z = f(p) function: both z=f(p) and z=f(P) equations are the M equation with P-A=p, A>0. For all the p+A=s, r=p<P, z variables we obtain (s is variable in the third system of coordinates with the (-A,0) center). With help of the second theme in some other new system of coordinates the M equations is a x-field of motions in relation to some regular function (z=f(-x+iy) is the x-field of motions in relation to the z=f(x+iy) function). It is proved, that the M equation is possible to be considered as regular function with point of view of other mathematical facts in the same new the system of coordinates: in the system of coordinates instead of the OX axis we consider the iOX axis, and instead of the iOY axis we consider the OY axis. The M set of points is fixed.

We consider the surprising regularity   of the transform of Laplace from the transform of Fourier from the point of view of the odd or even reflection of the transform   in relation to the center of co-ordinates   too.  As result, the transform of Laplace from the transform of Fourier of S(x) is odd or even in usual assumption about the S(x) function.