Download PDF by M. H. Lantsman (auth.): Asymptotics of Linear Differential Equations

By M. H. Lantsman (auth.)

ISBN-10: 9048157730

ISBN-13: 9789048157730

ISBN-10: 9401597979

ISBN-13: 9789401597975

The asymptotic thought offers with the problern of choosing the behaviour of a functionality in a local of its singular aspect. The functionality is changed by means of one other identified functionality ( named the asymptotic functionality) shut (in a feeling) to the functionality into consideration. Many difficulties of arithmetic, physics, and different divisions of average sci­ ence carry out the need of fixing such difficulties. today asymptotic idea has develop into a tremendous and autonomous department of mathematical research. the current attention is principally according to the speculation of asymp­ totic areas. each one asymptotic area is a suite of asymptotics united by way of an linked genuine functionality which determines their progress close to the given aspect and (perhaps) another analytic homes. the most contents of this publication is the asymptotic conception of normal linear differential equations with variable coefficients. The equations with strength order progress coefficients are thought of intimately. because the program of the speculation of differential asymptotic fields, we additionally think about the subsequent asymptotic difficulties: the behaviour of particular and implicit capabilities, mistaken integrals, integrals depending on a wide parameter, linear differential and distinction equations, and so forth .. The acquired effects have an autonomous that means. The reader is thought to be accustomed to a entire process the mathematical research studied, for example at mathematical departments of universities. extra helpful info is given during this publication in summarized shape with proofs of the most aspects.

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Let Zl' Z2 be arbitrary points of r. We denote the length of r (z 1 , z 2 ) by l (r (z1 , z2 )). 10. Consider an integral of the form I(z) = { f(s)ds. 12. Let r be anormal curve in the complex plane. 10) converge for any z E r. Let hr{f(z)} = k. Here zo = oo if k < -1, and zo is a sufficiently Zarge positive number if k ~ -1. Then hr{J{z)} ~ k + 1. Some other examples of asymptotic spaces will be given in the subsequent chapters. 4. SERIES AND SEQUENCES IN ASYMPTOTIC SPACES Consider a series of the form Oo + 01 + ...

5. 8. Let hv{f(z)} = p and hv{g(z)} = q. } = O"hv{f(z)}, where O" is a real non-negative number; (2) hv{f(z)+g(z)} :::; max{p, q}. If in addition there exists a finite limit lim z-+oo,zED g(z) =/= -1 f(z) then hv{f(z) + g(z)} = max{p, q}. (3) hv{f(z)g(z)} :::; hv{f(z)} + hv{g(z)}. So that Hv is an asymptotic ring. 9. Consider an algebraic equation of the form yn = a1(z)yn-l + a2(z)yn- 2 + ... 6) where aj(z) E Hv (j = 1, 2, ... , n). A(z)) of the equation belongs to H v. 7) PROOF. 2). Suppose the contrary.

9. NORMED SPACES An important particular case of a linear metric system is a linear normed space. In such spaces, we denote the length function by llxll which is called the norm of x. 40. Linear metric system ME is called a linear normed space if for any x E ME and any complex (real) number A E N, the following inequality holds: IIAxll IAIIIxll· So that a linear normed space satisfy the following properties: Let E = {x} be a linear system. Let, for any x, it be assigned a non-negative number llxll satisfying the following conditions (axioms): = (1) llxll = 0 if and only if x = 0; (2) for any x, y E E, llx + Yll ~ llxll + llzll (triangle axiom); (3) for any complex (real) number A, llAxll = IAIIIxllA complete normed space is also called a Banach space, or a space of type B.

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