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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)