I. Complex Algebra and Functions
1 Algebra of Complex Numbers

Complex Plane

Polar Form
2 cis(y) = exp(iy)


Geometric Series
3 Functions of Complex Variable

4 Cauchy-Riemann Conditions

Harmonic Functions
5 Simple Mappings: az+b, z2, √z

Idea of Conformality
6 Complex Exponential
7 Complex Trigonometric and Hyperbolic Functions
8 Complex Logarithm Problem set 1 due
9 Complex Powers

Inverse Trig. Functions
10 Broad Review ... Probably focusing on sin-1z
II. Complex Integration
11 Contour Integrals
12 Path Independence
Exam 1
13 Cauchy's Integral Theorem
14 Cauchy's Integral Formula

Higher Derivatives
15 Bounds

Liouville's Theorem

Maximum Modulus Principle
16 Mean-value Theorems

Fundamental Theorem of Algebra
17 Radius of Convergence of Taylor Series Problem set 2 due
III. Residue Calculus
18 Laurent Series
19 Poles

Essential Singularities

Point at Infinity
20 Residue Theorem

Integrals around Unit Circle
21 Real Integrals From -∞ to +∞

Conversion to cx Contours
22 Ditto ... including Trig. Functions

Jordan's Lemma
Exam 2
23 Singularity on Path of Integration

Principal Values
24 Integrals involving Multivalued Functions
IV. Conformal Mapping
25 Invariance of Laplace's Equation
26 Conformality again

Inversion Mappings
27 Bilinear/Mobius Transformations Problem set 3 due
28 Applications I
29 Applications II
V. Fourier Series and Transforms
30 Complex Fourier Series
31 Oscillating Systems

Periodic Functions
32 Questions of Convergence

Scanning Function

Gibbs Phenomenon
33 Toward Fourier Transforms
34 Applications of FTs
Exam 3
35 Special Topic: The Magic of FFTs I
36 Special Topic: The Magic of FFTs II
Final Exam