An Introduction to Complex Function Theory

Numerical testing of the stability of viscous shock waves

“…For example, calculation of D and D can be done directly giving more accurate values in equation (21), a higher order Runge-Kutta method could be used in the numerical integration for greater speed, or integrated equations [2] could be substituted to alleviate the problem of zeroes at the origin. But accuracy is already very good and would not likely improve significantly with implementation of these ideas.…”

“…2. Apply the argument principle [21] to Ω to find the number of zeroes of D in the closed right half plane. 3.…”

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of Möbius transformations

“…As a global analytic function (the definition of a global analytic function can be found in [18]), Ω(x, y) is, in general, a multiple-valued mapping (called also a set-valued mapping), which Differential Galois theory and solvable subgroup of Möbius transformations 751 has at most a countable number of single-valued branches, and every branch is analytic in a corresponding open domain dense inĈ 2 and satisfies…”

The fractal structure of limit set of solution space of a doubly periodic Ricatti equation

“…The As a global analytic function (see [7]), a solution of equation (1) is usually multiple-valued It is proved in [5] that (1) has two particular solutions z,. (t) and z,2(t), which have respectively single-valued and analytical periodic branches rl (t) and L2 (t) with the period 2cr in the strip region D {tlllm(t)l < },…”

“…It is well known that the limit sets of many nonlinear autonomous systems have fractal structures (see [1][2] [3]) However, it is very difficult to verify the fractal structure analytically and exactly for a given system, since the nonlinear autonomous system is usually not integrable by traditional quadratures In [4] and [5] the periodic solutions and the structures of Riemann surfaces of solutions to a doubly periodic Riccati equation are discussed, and the monodromy group (a Kleinian group (see [6])) of the equation is established concretely for the study of the structure of the limit set of the solution space, and it is conjectured that the limit set has a fractal structure under certain conditions In this paper, this conjecture will be proved by means of the Kleinian group Some As a global analytic function (see [7]), a solution of equation (1) The c-values of the local periodic solutions r, (t), r2 (t), "q (t) and ,2 (t) are respectively 0, c, and (see [5]) For given A E R+, let V(A) be the union of sets v() s(o)Us(oo)Us()Us(-).…”