# Readings

These chapters come from the text: Hildebrand, Francis.

*Advanced Calculus for Applications.*2nd ed. Englewood Cliffs: Prentice Hall, March 31, 1976. ISBN: 0130111899.LEC # | TOPICS | READINGS |
---|---|---|

1-2 | Number Systems and Algebra of Complex Numbers Elementary Complex Functions, Part 1 |
Algebra of Complex Numbers (10.1) Elementary Complex Functions (10.2, 10.3) |

3-4 | Elementary Complex Functions, Part 2 Branch Points and Branch Cuts |
Analytic Functions (10.3, 10.4) Convergence of Power Series: Examples (4.1, 4.2) Line Integrals, Cauchy's Formula (10.5, 10.6) |

5-6 | Analytic Functions Complex Integrals |
Taylor Series, Laurent Series (10.7, 10.8) Singularities (10.9) |

7-8 | Cauchy's Formula, Properties of Analytic Functions Taylor Series, Laurent Series |
Singularities at Infinity (10.10, 10.11) The Residue Theorem (10.12) |

9-10 | Laurent Series (cont.) Properties of Laurent Series, Singularities |
Evaluation of Real Definite Integrals (10.13) Handout 1 on Overview of Evaluation of (Real) Definite Integrals (PDF) Limiting Contours (10.14) Indented Contours (10.15) |

11-12 | Singularities (cont.) Residue Theorem |
Indented Contours (10.15) Integrals Involving Branch Points (10.16) |

14-15 | Evaluation of Real Definite Integrals, Case I Evaluation of Real Definite Integrals, Case II |
Singular Points of Linear Second-Order ODEs (4.3) The Method of Frobenius (4.4) Handout 2 on An Overview of the Fobenius Method (PDF) |

16-17 | Evaluation of Real Definite Integrals, Case III Evaluation of Real Definite Integrals, Case IV |
The Method of Frobenius - Exceptional Cases (4.4, 4.5, 4.6) |

18-19 | Theorems for Contour Integration Series and Convergence |
Bessel Functions (4.8) Properties of Bessel Functions (4.9) |

20-21 | Ordinary Differential Equations Singular Points of Linear Second-Order ODEs |
Differential Equations Satisfied by Bessel Functions (4.10) Legendre Functions (4.12) |

22-23 | Frobenius Method Frobenius Method - Examples |
Introduction to Boundary Value Problems (5.1) The Rotating String, The Rotating Shaft (5.2, 5.3) |

24-25 | Frobenius Method (cont.) and a "particular type" of ODE Bessel Functions |
Orthogonality of Characteristic Functions (5.6) Expansions in Series of Orthogonal Functions (5.7) |

26-27 | Properties of Bessel Functions Modified Bessel Functions |
Boundary Value Problems for Nonhomogeneous PDEs (5.8) Fourier Series (5.10) Complete Fourier Series (5.11) |

29-30 | Differential Equations Satisfied by Bessel Functions Introduction to Boundary-Value Problems |
Fourier-Bessel Series (5.13) Legendre Series (5.14) The Fourier Integral (5.15) |