Pavlov A.V. Reflection of Regular Functions
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https://doi.org/10.15688/mpcm.jvolsu.2021.4.6
Andrey V. Pavlov
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Higher Mathematics-1,
Moscow Institute of Radiotechnics, Electronics and Automatics —
Russian Technological University
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https://orcid.org/0000-0002-1082-2222
Prosp. Vernadskogo, 78, 117454 Moscow, Russian Federation
Abstract. 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=const. The F(p) field is the field of the moved functions.
Key words: regular function, double reflection, periodicity, even functions, moved functions, field of complex values.
Reflection of Regular Functions by Pavlov A.V. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 24 No. 4 2021, pp. 79-82
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