LEC #  TOPICS  KEY DATES 

I. Complex Algebra and Functions  
1  Algebra of Complex Numbers Complex Plane Polar Form 

2  cis(y) = exp(iy) Powers Geometric Series 

3  Functions of Complex Variable Analyticity 

4  CauchyRiemann Conditions Harmonic Functions 

5  Simple Mappings: az+b, z^{2}, √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^{1}z  
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  Meanvalue 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 