18.04 | Fall 1999 | Undergraduate

# Complex Variables with Applications

## Calendar

Week # Lec # Topics Key Dates
1 1 1.1 Complex Algebra. Complex Plane. Motivation and History
2 1.2 Polar Form. Complex Exponential. DeMoivre’s Theorem
3 1.3 Newton’s Method. Fractals
2-3 4 2.1 Complex Functions. Analyticity
5 2.2 Cauchy-Riemann Eqns. Harmonic Functions
6 2.3 Exponential and Trig. Functions. Logarithmic Function
7 2.4 Branch Cuts. Applications
8 2.5 Complex Powers and Inverse Trig. Functions
4-5 9 3.1 Contour Integrals
10 3.2 Path Independence
11 3.3 Cauchy’s Theorem
12 3.4 Cauchy’s Integral Formula
13 3.5 Liuville’s Theorem. Mean Value and Max. Modulus
14 3.6 Dirichlet Problem
15   EXAM #1: covering 1, 2 and about 1/2 of 3.
6 16 4.1 Taylor Series. Radius of Convergence
17 4.2 Laurent Series
18 4.3 Zeros. Singularities. Point at Infinity
7-8 19 5.1 Residue Theorem. Integrals over the Unit Circle
20 5.2 Real Integrals. Conversion to Complex Contours
21 5.3 Trig. Integrals. Jordan’s Lemma
22 5.4 Indented Contours. Principal Value
23 5.5 Integrals Involving Multi-valued Functions
24 5.6 Argument Principle and Rouche’s Theorem
25   EXAM #2: covering second half of 3, 4 and 5.
9-10 26 6.1 Complex Fourier Series
27 6.2 Oscillating Systems. Periodic Functions
28 6.3 Applications of Fourier Series
29 6.4 Fourier Transform and Applications
30 6.5 Laplace Transform and Inversion Formula
11-12 31 7.1 Invariance of Laplace’s Eqn. Conformality
32 7.2 Inversion Mapping. Bilinear Mappings
33 7.3 Examples and Applications
34 7.4 More Examples (if time permits)
35   EXAM #3: covering 6 and 7.

## Course Info

Fall 1999
##### Learning Resource Types
Exams with Solutions
Problem Sets with Solutions