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Complex Variables with Applications >> Content Detail



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LEC #TOPICSKEY DATES
I. Complex Algebra and Functions
1Algebra of Complex Numbers

Complex Plane

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

Powers

Geometric Series
3Functions of Complex Variable

Analyticity
4Cauchy-Riemann Conditions

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

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

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

Higher Derivatives
15Bounds

Liouville's Theorem

Maximum Modulus Principle
16Mean-value Theorems

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

Essential Singularities

Point at Infinity
20Residue Theorem

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

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

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

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

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

Periodic Functions
32Questions of Convergence

Scanning Function

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

 








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