A Review of Complex Variable Theory

Some of the most frequently encountered and generic types of plane vector fields with singular points at the origin of the coordinate system have been studied using complex variable theory methods combined with complex potential methods and field theory methods. The basic concepts of field theory and vector analysis, which are used to study vector fields and the main numerical characteristics of these fields, have been considered. The study of the most frequently encountered vector fields with singular points of four types, namely the generator, the vortical point, the eddy source, the dipole, have been conducted. The application of the complex potential for finding the main characteristics of the vector fields of the considered types, namely their divergence and rotor, has been shown. Equipotential lines and streamlines of the considered vector fields have been obtained and graphically constructed using the method of complex potential. Studied using the vector analysis methods and the methods of the theory of complex variable functions (complex potential) characteristics of vector fields can be used for mathematical modeling of various problems, arising during the study of layers, namely soil and water reservoir filtration problems, as well as in studying the flow of fluid or gas in layers problems. The developed and considered mathematical models of flat vector fields and the found numerical characteristics of these fields can be used to solve other problems of the oil and gas complex, which require studies of the flow of liquids or gases in gas- or oil-bearing beds.

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Integrability of Motions in Galactic Potentials

Publications of the Astronomical Society of Australia ◽

10.1017/s1323358000018361 ◽

1986 ◽

Vol 6 (4) ◽

pp. 453-458

Author(s):

Paul Cleary

Keyword(s):

Angular Momentum ◽

Complex Variable ◽

Painlevé Analysis ◽

Variable Theory ◽

Complex Variable Theory ◽

Galactic Potentials ◽

Full Complement ◽

Painleve Analysis

AbstractThe dynamics exhibited by systems, such as galaxies, are dominated by the isolating integrals of the motion. The most common are the energy and angular momentum integrals. The motions in a system with a full complement of isolating integrals are regular, that is, periodic or quasi-periodic. Such a system is integrable. If there is a deficiency in the number of integrals, then the motions are chaotic. There is a fundamental quantative difference in the motion, depending on the number of integrals. A technique, called Generalised Painlevé analysis, based on complex variable theory allows the user to determine if a system is integrable. Two new integrable cases of the Henon-Heiles system are presented, bringing the total number of such integrable potentials to five. It is highly probable that there are no further integrable cases of the Henon-Heiles potential. Five cases of the quartic Verhulst potential, defined by certain restrictions on the coefficients, which are found to be integrable are summarised.

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