week | Material covered so far | Exercises (Solutions written by Leslie) | (To hand in on week n+1) |
6 | The e z and log(z) functions.
Analytic functions. Holomorphic functions. The Cauchy Riemann equations. Angles under holomorphic maps. |
I§1: 4, 5, 7, 8. I§2: 2, 8, 10 | |
7 | Convergence radius. Analyticity of functions defined via power series. Differentiation of power series. Analytic ⇒ Holomorphic. |
I§2: 3, 4, 11, 12. I§3: 1, 2, 3, 4. | |
8 | Formulas for the convergence radius (p. 55 & 57). The set of zeroes of an analyticity function is discrete. Analytic continuation. |
I§4: 1, 3, 5. II§1: 1, 3, 4. II§2: 1, 3, 4, 5, 6, 10. | |
9 | Inverse function of an analytic function (p. 76 - 79). My proof used Holomorphic ⇒ Analytic, which we haven't shown yet. |
II§1: 6, Prove the cases 0 and ∞ of Thm 2.6 (p. 55). II§2: 7, 11, 12, 13, II§4: 1, 2. |
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10 | Open mapping theorem. Maximum modulus principle. Integrals along paths. | II§3: 4, 5 II§5: 1, 2, 3, 4, 5, 6 II§6: 1 - 5 | |
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12 | Invariance of the integral under homotopies of paths and of loops. The concept of simply connected topological space. |
III§1: 2, 3. III§2: 1, 3, 4, 8, 10, 11. | |
13 | Winding number. Existence of local primitives. Holomorphic ⇒ Analytic. | III§5: 1, 2, 3, 4. III§6: 1, 2, 3, 4, 5, 6, 7, 8. | |
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15 | Poles and residues. The residue theorem (see pages 173,174, and 144). | III§7: 1, 3. V§2: 6, 11, 13. VI§1: 1, 3, 5, 7, 9, 11, 13, 19, 20. |
the proof of Corollary 7.6*. |
16 | have written Theorem 7.7... You may chose which one you hand in: either 7.6 or 7.7. |
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17 | Proof of the residue theorem (p. 143-154). |
V§1: 1, 2. V§2: 1, 2, 3, 4, 5, 8. | |
18 | Existence of Laurent expansions. | V§1: 6, 7. V§2: 7, 9, 10, 12, 15. | |
19 | Uniform limits of analytic functions. The function Σn ∈ ℤ (z-n)-1. Liouville's thm. | V§1: 3, 4, 5. V§3: 1, 2, 3, 4, 5. | |
20 | Computation of definite integrals. The example of ∫-∞∞ cos(πx/2)/(1+x4) dx. | V§3: 9, 10, 11. VI§2: 1, 2, 3, 4. | |
21 | More definite integrals: the Mellin transofrm of 1/(1+x2).
The Riemann sphere ℂℙ1. Classification of holomorphic maps ℂℙ1 → ℂℙ1. Fractional linear transofrmations. |
VI§2: 5, 6, 7, 8, 9, 11, 17, 18, 19, 20. | |
22 | Automorphisms of the Riemann sphere. Analytic automorphisms of the upper half plane and of the unit disc. Statement of the Riemann mapping theorem. |
VII§5:1, 2, 3, 4, 5, 6. | |
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24 |