## Asymptotics of Linear Differential EquationsThe asymptotic theory deals with the problern of determining the behaviour of a function in a neighborhood of its singular point. The function is replaced by another known function ( named the asymptotic function) close (in a sense) to the function under consideration. Many problems of mathematics, physics, and other divisions of natural sci ence bring out the necessity of solving such problems. At the present time asymptotic theory has become an important and independent branch of mathematical analysis. The present consideration is mainly based on the theory of asymp totic spaces. Each asymptotic space is a collection of asymptotics united by an associated real function which determines their growth near the given point and (perhaps) some other analytic properties. The main contents of this book is the asymptotic theory of ordinary linear differential equations with variable coefficients. The equations with power order growth coefficients are considered in detail. As the application of the theory of differential asymptotic fields, we also consider the following asymptotic problems: the behaviour of explicit and implicit functions, improper integrals, integrals dependent on a large parameter, linear differential and difference equations, etc .. The obtained results have an independent meaning. The reader is assumed to be familiar with a comprehensive course of the mathematical analysis studied, for instance at mathematical departments of universities. Further necessary information is given in this book in summarized form with proofs of the main aspects. |

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### Contents

II | 1 |

III | 5 |

IV | 9 |

V | 11 |

VI | 14 |

VII | 16 |

VIII | 17 |

IX | 22 |

XLVII | 183 |

XLVIII | 193 |

XLIX | 200 |

L | 204 |

LI | 206 |

LII | 214 |

LIII | 224 |

LIV | 230 |

X | 23 |

XI | 24 |

XII | 26 |

XIII | 27 |

XIV | 29 |

XV | 31 |

XVI | 34 |

XVII | 38 |

XVIII | 39 |

XIX | 40 |

XX | 43 |

XXI | 44 |

XXII | 46 |

XXIII | 49 |

XXIV | 50 |

XXV | 54 |

XXVI | 56 |

XXVII | 63 |

XXVIII | 67 |

XXIX | 71 |

XXX | 84 |

XXXI | 90 |

XXXII | 96 |

XXXIII | 103 |

XXXIV | 106 |

XXXV | 113 |

XXXVI | 116 |

XXXVII | 119 |

XXXVIII | 126 |

XXXIX | 128 |

XL | 133 |

XLI | 137 |

XLII | 144 |

XLIII | 147 |

XLIV | 153 |

XLV | 166 |

XLVI | 171 |

LV | 233 |

LVI | 247 |

LVII | 256 |

LVIII | 258 |

LIX | 270 |

LX | 273 |

LXI | 274 |

LXII | 276 |

LXIII | 282 |

LXIV | 295 |

LXV | 307 |

LXVI | 308 |

LXVII | 314 |

LXVIII | 320 |

LXIX | 332 |

LXX | 335 |

LXXI | 340 |

LXXII | 346 |

LXXIII | 349 |

LXXIV | 354 |

LXXV | 355 |

LXXVI | 365 |

LXXVII | 371 |

LXXVIII | 374 |

LXXIX | 378 |

LXXX | 380 |

LXXXI | 392 |

LXXXII | 398 |

LXXXIII | 402 |

LXXXIV | 411 |

LXXXV | 413 |

LXXXVI | 417 |

LXXXVII | 420 |

LXXXVIII | 425 |

437 | |

438 | |

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### Common terms and phrases

ai(t Airy equation algebraic algebraic extension an(t analytic asymptotic limit asymptotic separability asymptotic series asymptotic solutions asymptotic space belonging to Q characteristic equation characteristic polynomial Clearly complete set complex numbers complex plane continuous function contractive mappings convergent Definition denote difference equations domain easy to show element equal equivalent estimate Example exists a number field of type field Q fixed formal solution function f(t fundamental matrix gi(t Hence holomorphic function induction with respect inequality integral interval large positive number last equation leads Lemma Let us substitute Let X(t linear differential equations linearly independent Math matrix metric space Moreover polynomial positive semi-axis possesses the property PROOF property of asymptotic prove r+oo real number required property root of equation saddle point sequence sm(t set of roots simple consequence singular solution to equation solution x(t sufficiently large positive suppose taking into account tion Xi(t xn(t zero