| LEC # | TOPICS | KEY DATES |
|---|---|---|
| I. Complex Algebra and Functions | ||
| 1 | Algebra of Complex Numbers Complex Plane Polar Form |
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| 2 | cis(y) = exp(iy) Powers Geometric Series |
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| 3 | Functions of Complex Variable Analyticity |
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| 4 | Cauchy-Riemann Conditions Harmonic Functions |
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| 5 | Simple Mappings: az+b, z2, √z Idea of Conformality |
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| 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 |
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| 15 | Bounds Liouville's Theorem Maximum Modulus Principle |
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| 16 | Mean-value Theorems Fundamental Theorem of Algebra |
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| 17 | Radius of Convergence of Taylor Series | Problem set 2 due |
| III. Residue Calculus | ||
| 18 | Laurent Series | |
| 19 | Poles Essential Singularities Point at Infinity |
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| 20 | Residue Theorem Integrals around Unit Circle |
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| 21 | Real Integrals From -∞ to +∞ Conversion to cx Contours |
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| 22 | Ditto ... including Trig. Functions Jordan's Lemma |
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| Exam 2 | ||
| 23 | Singularity on Path of Integration Principal Values |
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| 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 |
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| 32 | Questions of Convergence Scanning Function Gibbs Phenomenon |
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| 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 | ||