## Advanced Calculus: An Introduction to Modern AnalysisIntroduction to linear algebra and ordinary differential equations; Limits and metric spaces; Continuity, compactness and connectedness; The derivative: theory and elementary applications; A first look at integration; Differentiation of functions of several variables; An introduction to fourier analysis; An introduction to modern integration theory; An introduction to complex integration. |

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### Contents

Preliminaries | 1 |

a A a is a member of the set A | 5 |

I B complement of B relative to A | 11 |

complex plane | 18 |

1ntroduction to Linear Algebra | 27 |

jC norm or length of x | 37 |

PA B conditional probability | 43 |

Limits and Metric Spaces | 67 |

Differentiation of Functions of Several Variables | 187 |

Sequences and Series | 249 |

Elementary Applications of 1nfinite Series | 293 |

An 1ntroduction to Fourier Analysis | 329 |

An 1ntroduction to Modern 1ntegration Theory | 395 |

An lntroduction to Complex 1ntegration | 465 |

The Fourier and Laplace Transforms | 537 |

A Sampling of Numerical Analysis | 589 |

Continuity Compactness and Connectedness | 91 |

The Derivative Theory and Elementary Applications | 125 |

A First Look at 1ntegration | 147 |

Answers to Selected Problems | 647 |

Table of Laplace Transforms | 662 |

### Common terms and phrases

1t follows adherence point analytic approximation arbitrary bounded calculation Cauchy sequence Chapter closed cluster point compact complex number Consequently constant continuous function continuously differentiable contraction map converges uniformly countable Definition Suppose denote determined differential equation equal example exists Figure finite number Fourier series Furthermore given graph hence Hint if/is integer interval a,b Lebesgue limit linear transformation mathematical mean value theorem n e Z+ neighborhood Newton's method Note Observation obtain open set partial derivatives partial sums path polynomial positive integer positive number problem R1 is continuous R1 is defined rational numbers reader real number rectifiable path result Riemann integrable satisfies Section sequence xn solution step functions summable function Suppose that X,d Taylor series that/is then/is Theorem 1f Theorem Suppose tion unique vector space x e a,b zero