Geometric and physical meaning of the function of real and complex variables

About the author:

Alexander D. Novikov, Cand. Sci. (Pedag.), Associate Professor, Department of Mathematics and its Teaching Methods, Armavir State Pedagogical Academy

Abstract:

This article analyzes the various approaches to the theory of functions of a complex variable to identify its geometric meaning. Different interpretations of the geometric meaning of the functions of a complex variable are considered. Using a geometric interpretation of the function of a complex variable in the form of mapping one complex to another and relevant analogy for extending the functions of one real variable, its physical meaning is revealed. Physical meaning of the function of a complex variable is identified by the method of analogies. Thus, the interpretation of the physical meaning of the functions of real and complex variables, based on a common approach to it (meaning) understanding as to the real function, and for the function of a complex variable is proposed. The deformation coefficient domain of the function at the point is used as a quantitative evaluation of the functions properties. It is a quantitative measure of the change in the density of uniformly distributed points in the given map. In accordance with the deformation pattern (tension, compression), this ratio is ether less or higher than neutral element.

References:

1. Luzin, N.N. Integral'noe ischislenie [Integral calculus]. Moscow, 1958. 415 p. (in Russian).

2. Shabat, B.V. Vvedenie v kompleksnyi analiz [Introduction to complex analysis]. Moscow, 1969. 576 p. (in Russian).3. Uitteker, E.E., Vatson, Dzh.N. Kurs sovremennogo analiza [A course of modern analysis] / Transl. fr. Eng. Moscow, 2002. 856 p. (in Russian).

4. Goncharov, V.L. Teoriia funktsii kompleksnogo peremennogo [Theory of functions of a complex variable]. Moscow, 1955. 352 p. (in Russian).

5. Khrestomatiia po istorii. Matematicheskii analiz. Teoriia veroiatnostei. Posobie dlia studentov ped. in-tov [Readings on the history. Mathematical analysis. The theory of probabilities. Students’ manual of pedagogical institutions] / Ed. by A.P. Iushkevich. Moscow, 1977. 224 p. (in Russian).

6. Lavrent'ev, M.A. Konformnye otobrazheniia [Conformal mappings]. Moscow, 1946.159 p. (in Russian).

7. Solomentsev, E.D. Funktsii kompleksnogo peremennogo i ikh primeneniia [Functions of a complex variable and their applicability]. Moscow, 1988. 197 p. (in Russian).

8. Khaplanov, M.G. Teoriia funktsii kompleksnogo peremennogo [Theory of functions of a complex variable]. Moscow, 1965. 208 p. (in Russian).

9. Gorin, E.A. Vvedenie v teoriiu analiticheskikh funktsii [Introduction to the theory of analytic functions]. Moscow, 2005. 163 p. (in Russian).

10. Markushevich, A.I., Markushevich, L.A. Vvedenie v teoriiu analiticheskikh funktsii [Introduction to the theory of analytic functions]. Moscow, 1977. 320 p. (in Russian).