It is proved in article, that a function with reflection in
relation to some point can be the double reflection of initial
function in relation to some other
points. The double reflection results in the periodicity of some
analytical function. In a
example we obtain a periodic odd function, if we move the result of two reflections.
We obtain a similar result after consideration of the $F(p)$ field:
$F(p)=f(p-2A)$, if $p=x+iy$, $x=A$, for all $A$. The $F(A+B+iy)$ values
equal to the $f(z-2A-2B)$ values in the $A+B+iy$ point as a result
of two moving of the $f(p)$ function to the right for all $y$ (at
first we move the $f(-A+iy$) on the $2A$ distance, after we move the
the $F(p)=f(A+iy)=f(p)$ function on the $2B$ distance in relation
to center in the $(A,0)$ point); the result of the such double
moving is equal to the values of initial field in the $A+B+iy$
point. The reflection of the $F(p)$ field in the $(A,0)$ and $(A+B,0)$
points is the $f(-p)$ regular function for all real $A$, $B$. We can
move the $F(p)$ values in reverse direction (to the left). In
the situation the values (in the left part of plane) are equal to
the values of initial regular $f(p)$ function. As a result of two
moving we obtain a new $G(p)$ field in relation to the $f(p)$
function after the movements to the left with the $(-A,0)$ center.
The regular $f(p)$ function is equal to the $G(p)$ field, if $ p-A$. It
is proved, that the $f(p)$ function is periodic.
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). If the $f(p)$ function
is regular in the left half of plane, the fact results in the
equality $f(p)=c$ too, $c=\mathrm{const}$. The $F(p)$ field is the field of the
moved functions.
