Math Notes

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Math Notes

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Complex Analysis

Part 1 - Complex Differentiation

1. Definition 2.1.1 - Complex differentiability

2. Remark 2.1.1

3. Theorem 2.1.1 - Cauchy-Riemann

4. Some examples

5. Definition 2.1.2 - Holomorphic

6. Corollary 2.1.1

7. Proposition 2.1.1

8. Corollary 2.1.2

9. Wirtinger Calculus

10. Theorem 2.1.1

11. Lemma 2.1.21- Cauchy Integral Theorem for rectangles

12. Theorem 2.1.2 - Cauchy's theorem for C1 images of rectangles.

13. Some applications

14. Theorem 2.1.3 - Integral Formula of Cauchy

15. Theorem 2.1.4 - power series expansion

17. Corollary 2.1.3 - Theorem of Goursat

18. Corollary 2.1.4 - Cauchy Estimate for C_n

19. Corollary 2.1.5 - Theorem of Louiville

20. Corollary 2.1.6 - Fundemental Theorem of Algebra

21. Definition 2.1.3 - Domains

22. Theorem 2.1.5 - Uniqueness Theorem

Part 2 - Zeros of Infinite Order

1. Definition 2.2.1

2. Proposition 2.2.1

3. Definition 2.2.2

4. Theorem 2.2.1

5. Theorem 2.2.2

6. Theorem 2.2.3 - Open Mapping Theorem

7. Corollary 2.2.1 - the maximum principle

8. Theorem 2.2.4 - Schwarz Lemma

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Let $D \subseteq C$ be a domain, $f : D \to C$ be holomorphic and non-constant. Then $f (D)$ is 1;; a domain.

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7. Corollary 2.2.1 - the maximum principle

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