| 1 |
1 |
1.1 Complex Algebra. Complex Plane. Motivation and History |
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2 |
1.2 Polar Form. Complex Exponential. DeMoivre's Theorem |
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3 |
1.3 Newton's Method. Fractals |
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| 2-3 |
4 |
2.1 Complex Functions. Analyticity |
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5 |
2.2 Cauchy-Riemann Eqns. Harmonic Functions |
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6 |
2.3 Exponential and Trig. Functions. Logarithmic Function |
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7 |
2.4 Branch Cuts. Applications |
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8 |
2.5 Complex Powers and Inverse Trig. Functions |
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| 4-5 |
9 |
3.1 Contour Integrals |
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10 |
3.2 Path Independence |
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11 |
3.3 Cauchy's Theorem |
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12 |
3.4 Cauchy's Integral Formula |
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13 |
3.5 Liuville's Theorem. Mean Value and Max. Modulus |
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14 |
3.6 Dirichlet Problem |
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15 |
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EXAM #1: covering 1, 2 and about 1/2 of 3. |
| 6 |
16 |
4.1 Taylor Series. Radius of Convergence |
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17 |
4.2 Laurent Series |
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18 |
4.3 Zeros. Singularities. Point at Infinity |
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| 7-8 |
19 |
5.1 Residue Theorem. Integrals over the Unit Circle |
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20 |
5.2 Real Integrals. Conversion to Complex Contours |
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21 |
5.3 Trig. Integrals. Jordan's Lemma |
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22 |
5.4 Indented Contours. Principal Value |
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23 |
5.5 Integrals Involving Multi-valued Functions |
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24 |
5.6 Argument Principle and Rouche's Theorem |
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25 |
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EXAM #2: covering second half of 3, 4 and 5. |
| 9-10 |
26 |
6.1 Complex Fourier Series |
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27 |
6.2 Oscillating Systems. Periodic Functions |
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28 |
6.3 Applications of Fourier Series |
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29 |
6.4 Fourier Transform and Applications |
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30 |
6.5 Laplace Transform and Inversion Formula |
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| 11-12 |
31 |
7.1 Invariance of Laplace's Eqn. Conformality |
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32 |
7.2 Inversion Mapping. Bilinear Mappings |
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33 |
7.3 Examples and Applications |
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34 |
7.4 More Examples (if time permits) |
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35 |
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EXAM #3: covering 6 and 7. |
