## Asymptotic Expansions for Ordinary Differential Equations"A book of great value . . . it should have a profound influence upon future research."-- Mathematical Reviews. Hardcover edition. The foundations of the study of asymptotic series in the theory of differential equations were laid by Poincaré in the late 19th century, but it was not until the middle of this century that it became apparent how essential asymptotic series are to understanding the solutions of ordinary differential equations. Moreover, they have come to be seen as crucial to such areas of applied mathematics as quantum mechanics, viscous flows, elasticity, electromagnetic theory, electronics, and astrophysics. In this outstanding text, the first book devoted exclusively to the subject, the author concentrates on the mathematical ideas underlying the various asymptotic methods; however, asymptotic methods for differential equations are included only if they lead to full, infinite expansions. Unabridged Dover republication of the edition published by Robert E. Krieger Publishing Company, Huntington, N.Y., 1976, a corrected, slightly enlarged reprint of the original edition published by Interscience Publishers, New York, 1965. 12 illustrations. Preface. 2 bibliographies. Appendix. Index. |

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### Contents

Some Basic Properties of Linear Differential Equations in the Complex Domain | 1 |

The Basic Existence Theorem and its Consequences | 3 |

Circuit Relations About Singular Points | 9 |

Regular Singular Points | 17 |

Solutions at a Regular Singular Point | 20 |

Asymptotic Power Series | 30 |

Definition of Asymptotic Power Series | 31 |

Elementary Propertles of Asymptotic Series | 33 |

Analytic Simplification | 143 |

Proof of Theorem 26 1 | 147 |

Shearing Transformations | 151 |

Turning Point Problems | 157 |

Analytic Theory | 169 |

Short Report on Other Turning Point Problems | 185 |

Nonlinear Equations | 197 |

Solution by Asymptotic Power Series | 200 |

The Existence of Asymptotic Series | 39 |

Irregular Singular Points | 49 |

Formal Simplification | 52 |

Analytic Simplification and Asymptotic Solution | 55 |

Miscellaneous Remarks | 61 |

Proof of the Main Asymptotic Existence Theorem when all Eigenvalues are Distinct | 65 |

The Stokes Phenomenon | 76 |

Generalizations by Means of Jordans Canonical Form | 88 |

General Case | 94 |

General Case | 99 |

General Case | 100 |

Some Special Asymptotic Methods | 116 |

Calculating Asymptotic Expansions from Con | 117 |

vergent Power Series | 122 |

Solution by Laplace Contour Integrals | 123 |

The Saddlepoint Method | 127 |

Asymptotic Expansions with Respect to a Parameter | 134 |

Formal Theory | 137 |

Transformation into a Linear Differential Equation | 202 |

Solution by Exponential Series | 213 |

Nonlinear Equations with a Parameter | 217 |

Singular Perturbations | 228 |

The Method of Visik and Lyusternik | 237 |

Qualitative Theory | 249 |

Series Expansions for the Initial Value Problem | 260 |

Nonlinear TwoPoint Boundary Value Problems | 279 |

Decomposition of General Linear Systems of Singular Perturbation Type | 287 |

General Remarks | 299 |

Linear Theory | 305 |

Series Expansions for Periodic Solutions of Singular Perturbation Problems | 315 |

Integration of Differential Equations by Factorial Series | 325 |

A Brief Summary of Some Recent Research | 347 |

353 | |

369 | |

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### Common terms and phrases

analytic function assume Assumption asymptotic expansion asymptotic power series asymptotic representation asymptotic series asymptotic solution boundary conditions boundary layer boundary value problem bounded calculated column components convergent corresponding defined denote depends determined diagonal matrix differen eigenvalues entries exists exponential factorial series follows formal power series formal series formula fundamental matrix fundamental matrix solution fundamental system Hence holomorphic hypotheses implies independent inequality infinity integral equation invariant factors leading matrix leading term left member Lemma linear differential equations Lyusternik matrix function method multiple neighborhood nonlinear nonsingular notation obtained parameter path of integration periodic solution polynomial possesses proof of Theorem properties proved recursive reduced problem region regular singular point relation right member satisfied scalar differential equation Section sector series expansion series in powers shearing transformation shifting matrix Sibuya singular perturbation solved subsector theory tial equation uniformly valid variable vector function Wasow x-plane

### Popular passages

Page 365 - FWJ Olver, Error bounds for asymptotic expansions, with an application to cylinder functions of large argument, Asymptotic Solutions of Differential Equations and their Applications, edited by CH Wilcox, ( New York, John Wiley and Sons, 1964).

Page 360 - Reduction of the order of a linear ordinary differential equation containing a small parameter,

Page 360 - Asymptotic theory of second order differential equations with two simple turning points,

Page 362 - The solutions of second order ordinary differential equations about a turning point of order two, Trans.

Page 363 - Approximate solution of a system of ordinary differential equations with a small parameter multiplying the derivatives.