Asymptotics of Linear Differential Equations Book

	Preface
Ch. 1	Introduction	1
1	Introduction	1
2	Conventions and notation	5
3	Asymptotic relations	9
Ch. 2	Metric spaces	14
1	Main definitions	14
2	Several examples of metric spaces	16
3	Sequences in a metric space	17
4	Equations in a metric space	22
5	Abstract functions	23
6	Linear sets	24
7	Metric groups. Main asymptotic relations in a metric group	26
8	Linear systems	27
9	Normed spaces	29
10	Basic properties of definite integrals	31
11	Solution of differential equations	34
Ch. 3	Asymptotic spaces	38
1	Definition of an asymptotic space	38
2	Examples of asymptotic spaces	39
3	Space of type H[subscript D]	40
4	Series and sequences in asymptotic spaces	43
5	Formal and asymptotic solutions	44
6	Factor Spaces	46
Ch. 4	Asymptotic behaviour of functions	49
1	Infinitesimal functions of a real argument	50
2	Analytic estimates	54
3	Series and sequences in the space II	56
4	Operators in the space II	63
5	Derivative estimates	67
6	Asymptotic behaviour of analytic functions	71
7	Implicit functions	84
Ch. 5	Power order growth functions on the positive semi-axis	90
1	Regularly varying functions	90
2	Class A[subscript t]	96
3	Subsets of the class A[subscript t] with some algebraic and analytic properties	103
4	Algebraic extensions of fields of type N[subscript S]	106
5	Algebraic extensions by roots of an algebraic system	113
6	Non-algebraic extensions	116
7	Estimates of roots of algebraic equations with comparable coefficients	119
8	Operators in a field of type N	126
Ch. 6	Power order growth functions of the complex argument	128
1	Classes A[subscript S] and A[subscript [epsilon]]
2	Subsets of class A[subscript S] with some algebraic and analytic properties	133
3	Power-logarithmic functions	137
Ch. 7	Integrals	144
1	Improper integrals	144
2	Integrals with a variable limit	147
3	The Laplace transformation	153
4	The Laplace method	166
5	The saddle point method	171
Ch. 8	Linear differential equations	183
1	Systems of linear differential equations	183
2	Single linear differential equations of nth order	193
3	Substitution x[prime] = yx	200
4	System of linear differential equations in the complex plane	204
Ch. 9	General asymptotic properties of linear differential equations	206
1	General estimates	206
2	The case when the coefficients rapidly tend to zero	211
3	Almost diagonal and L-diagonal systems	214
4	Single linear equation with asymptotically simple roots of the characteristic polynomial	224
Ch. 10	Linear differential equations with power order growth coefficients on the positive semi-axis	230
1	Formal solutions to a single linear differential equation of nth order	233
2	Asymptotic solutions to a single linear equation of nth order	247
3	Qualitative characteristics of linear differential equations	256
4	Second order differential equations	258
5	Systems of two differential equations with two unknowns	270
Ch. 11	Linear differential equations in singular cases on the positive semi-axis	273
1	Asymptotically independent functions	274
2	Formal asymptotic solutions	276
3	Connection between characteristic roots and solutions of a linear differential equation	282
4	Analytic transformations	295
Ch. 12	Linear differential equations in a sector of the complex plane	307
1	Formal solutions	308
2	Some auxiliary propositions	314
3	Main Theorems	320
4	Qualitative characteristics of linear differential equations in a sector of the plane	332
Ch. 13	Linear differential equations with power-logarithmic coefficients	335
1	Some auxiliary propositions	335
2	Main theorems	340
3	Deficiency indices of certain differential operators	346
4	Systems of linear differential equations	349
Ch. 14	Linear difference equations. General theory	354
1	First order equations	355
2	Systems of nth order equations	365
3	Single linear difference equations of nth order	371
4	Difference equations with constant coefficients	374
Ch. 15	Asymptotic behaviour of solutions of linear difference equations	378
1	The Poincare and Perron theorems	380
2	Linear difference equations with power order growth coefficients	392
3	Asymptotic solution to linear difference equation	398
4	Second order equations	402
5	Systems of two equations	411
6	Higher order equations	413
7	Differential - difference equations	417
Ch. 16	Supplement	420
1	Algorithms with an uniformly distributed theoretical error	420
	Bibliography	425
	List of symbols	437
	Index	438