## Real Analysis and Applications: Including Fourier Series and the Calculus of VariationsReal Analysis and Applications starts with a streamlined, but complete, approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]." The text not only provides clear, logical proofs, but also shows the student how to derive them. The excellent exercises come with select solutions in the back. This is a text that makes it possible to do the full theory and significant applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications along with the theory. The book is suitable for undergraduates interested in real analysis. |

### Contents

III | 3 |

IV | 9 |

V | 13 |

VI | 19 |

VIII | 23 |

X | 29 |

XII | 31 |

XIII | 33 |

XXXVIII | 101 |

XL | 105 |

XLII | 107 |

XLIII | 113 |

XLV | 117 |

XLVII | 119 |

XLIX | 121 |

LI | 127 |

XIV | 39 |

XVI | 43 |

XVIII | 45 |

XIX | 51 |

XXI | 57 |

XXII | 59 |

XXIII | 63 |

XXIV | 69 |

XXVI | 73 |

XXVIII | 79 |

XXX | 83 |

XXXII | 87 |

XXXIV | 91 |

XXXVI | 97 |

LV | 133 |

LVII | 137 |

LIX | 143 |

LXI | 147 |

LXIII | 151 |

LXV | 155 |

LXVI | 159 |

LXVIII | 163 |

LXX | 167 |

LXXII | 171 |

LXXIV | 179 |

LXXV | 191 |

LXXVI | 193 |

### Other editions - View all

Real Analysis and Applications: Including Fourier Series and the Calculus of ... Frank Morgan No preview available - 2005 |

### Common terms and phrases

accumulation point an/bn assume ball boundary point calculus of variations Cantor set Cauchy Chapter choose circle closed sets compact set Compute constant contained continuous function converges absolutely converges uniformly Corollary countable sets decimal place define definition derivative differential equation diverges domain dsē dx dy endpoints Euler's equation Exercise exp(x f is continuous Figure finite fn(x formula Fourier series fractal function f(x geodesics geometry give a counterexample Give an example given ɛ hence infinitely intersection lim fn limit maximum metric negative terms nonempty nonnegative open sets oscillation positive terms power series Proof Proposition radius of convergence rationals real analysis real numbers Riemann integral Riemann sums Riemannian sequence series converges absolutely Show sinē smooth solution sphere subintervals subset Suppose term by term totally disconnected U₁ uniformly continuous vanishes vector Weierstrass M-test yields მყ