| 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 |
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| 7 |
Complex Trigonometric and Hyperbolic Functions |
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| 8 |
Complex Logarithm |
Problem set 1 due |
| 9 |
Complex Powers
Inverse Trig. Functions |
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| 10 |
Broad Review ... Probably focusing on sin-1z |
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| II. Complex Integration |
| 11 |
Contour Integrals |
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| 12 |
Path Independence |
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Exam 1 |
| 13 |
Cauchy's Integral Theorem |
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| 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 |
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| 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 |
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| IV. Conformal Mapping |
| 25 |
Invariance of Laplace's Equation |
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| 26 |
Conformality again
Inversion Mappings |
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| 27 |
Bilinear/Mobius Transformations |
Problem set 3 due |
| 28 |
Applications I |
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| 29 |
Applications II |
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| V. Fourier Series and Transforms |
| 30 |
Complex Fourier Series |
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| 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 |
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| 34 |
Applications of FTs |
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Exam 3 |
| 35 |
Special Topic: The Magic of FFTs I |
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| 36 |
Special Topic: The Magic of FFTs II |
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Final Exam |
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