## Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation TheoryThis book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory and explains how to use these methods to obtain approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute-force numerical methods may not converge to useful solutions. The objective of this book is to teaching the insights and problem-solving skills that are most useful in solving mathematical problems arising in the course of modern research. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions; over 600 problems, of varying levels of difficulty; and an appendix summarizing the properties of special functions. |

### Contents

II | 3 |

III | 5 |

IV | 7 |

V | 11 |

VI | 14 |

VII | 20 |

VIII | 24 |

IX | 27 |

XLV | 276 |

XLVI | 280 |

XLVII | 302 |

XLVIII | 306 |

XLIX | 319 |

L | 324 |

LI | 330 |

LII | 335 |

X | 29 |

XI | 30 |

XII | 36 |

XIII | 37 |

XIV | 40 |

XV | 49 |

XVI | 53 |

XVIII | 61 |

XIX | 62 |

XX | 66 |

XXI | 68 |

XXII | 76 |

XXIII | 88 |

XXIV | 103 |

XXV | 107 |

XXVI | 118 |

XXVII | 136 |

XXVIII | 146 |

XXIX | 148 |

XXX | 152 |

XXXI | 171 |

XXXII | 185 |

XXXIII | 196 |

XXXIV | 205 |

XXXV | 206 |

XXXVI | 214 |

XXXVII | 218 |

XXXVIII | 227 |

XXXIX | 233 |

XL | 240 |

XLI | 247 |

XLII | 249 |

XLIII | 252 |

XLIV | 261 |

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### Common terms and phrases

a₁ a₂ Airy function an+1 analysis analytic approximation to y(x asymptotic analysis asymptotic behavior asymptotic expansion asymptotic matching asymptotic relation asymptotic series B₁ behavior of solutions behavior of y(x Bessel boundary conditions boundary layer boundary-value problem branch points c₁ c₂ coefficients compute constant contour critical point derive determine difference equation differential equation exact solution Example exponentially first-order formula Frobenius series full asymptotic gives initial conditions initial-value problem integral representation irregular singular point Laplace's method leading behavior leading-order linear local analysis nonlinear obtain optimal asymptotic approximation outer solution Padé approximants Padé sequence perturbation series perturbation theory plane polynomial power series Prob radius of convergence region result saddle point satisfies second-order Show solution y(x solve steepest-descent Substituting Taylor series tion trajectories valid values WKB approximation WKB theory y₁