By M. H. Lantsman (auth.)

The asymptotic idea offers with the problern of deciding on the behaviour of a functionality in a local of its singular element. The functionality is changed via one other identified functionality ( named the asymptotic functionality) shut (in a feeling) to the functionality into account. Many difficulties of arithmetic, physics, and different divisions of usual sci ence deliver out the need of fixing such difficulties. this day asymptotic conception has turn into an immense and autonomous department of mathematical research. the current attention is principally in response to the speculation of asymp totic areas. every one asymptotic area is a set of asymptotics united by way of an linked actual functionality which determines their progress close to the given aspect and (perhaps) another analytic houses. the most contents of this publication is the asymptotic thought of normal linear differential equations with variable coefficients. The equations with strength order progress coefficients are thought of intimately. because the software of the speculation of differential asymptotic fields, we additionally contemplate the subsequent asymptotic difficulties: the behaviour of specific and implicit capabilities, fallacious integrals, integrals depending on a wide parameter, linear differential and distinction equations, and so on .. The bought effects have an autonomous which means. The reader is thought to be accustomed to a entire process the mathematical research studied, for example at mathematical departments of universities. extra priceless info is given during this publication in summarized shape with proofs of the most aspects.

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**Example text**

Consider a limit of the form z = oo is its boundary point (we suppose that k = lim z-too,zED Here lim means an upper limit. 2) can be rewritten in the form lim sup ln lf(z)l_ T-t+oo lzi~T,zED ln lzl So that either k is a real number, or k = -oo, k = +oo. We denote by Hn a space of all complex functions such that if f(z) E Hn, then f(z) is defined for lzl » 1, z E D, and k < +oo. The value k is called a simple estimate of power growth (or simply a simple estimate). 3) means that f(z) E Hn and its simple estimate is equal to k.

14) if it isaformal solution to the equation, and there exists an exact solution y to the equation such that g :::::: y. 6. FACTOR SPACES The equation F(x) = -oo may have many solutions in EF. Therefore we introduce some moresimple spaces which (in definite sense) are equivalent to the corresponding asymptotic spaces. Let EF be an asymptotic space. 21. Let x be an arbitrary element of EF. Let F(x+O) = F(x) for any (} :::::: 0. We derrote by X the set consisting of all elements x* E EF such that x* :::::: x.

Xn)T ben-dimensional column-matrix where the Coordinatesare complex (real) numbers. Let B be a space consisting of column-matrices of the form X= (xl, ... ,xn)T and llxll = lxll + ... + lxnl· We set cB(a,ß) tobe a space of any continuous (column-matrices) functions x(t) = (x1(t), ... ,xn(t))T in J, and llx(t)llc = max lxl(t)l + ... + max lxn(t)l. tEJ tEJ Clearly, cB (a, ß) is a complete linear normed space. B is a space of all square complex (real) numerical matrices. For any A = (aij)n n IIAII = L .