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) |