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Math Notes

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Commutative Algebra

6. Chain Conditions

1. Proposition 4.6.1

2. Examples 4.6.1

3. Definition 4.6.1 - Noetherian or Artinian

4. Examples 4.6.2

5. Proposition 4.6.2

6. Proposition 4.6.3 - Noetherian respects exactness

7. Corollary 4.6.1 - noetherian direct sums

8. Definition 4.6.2 - Noetherian or Artinian Ring

9. Example 4.6.3 -Noetherian or Artinian Rings

10. Proposition 4.6.4 - finitely generated over noetherian

11. Proposition 4.6.5

7. Dimension of Modules

1. Definition 4.7.1 - length of a chain

2. Examples 4.7.1

3. Proposition 4.7.1 - properties of length

4. Proposition 4.7.2

5. Proposition 4.7.3 - length is additive

6. Proposition 4.7.4 - vector-space length

7. Corollary 4.7.1

8. Noetherian and Artinian Rings

1. Proposition 4.8.1 - homomorphic image of noetherian

2. Remark 4.8.1

3. Proposition 4.8.2

4. Theorem 4.8.1 - Hilbert's Basis Theorem

5. Corollary 4.8.1

6. Definition 4.8.1 - Krull Dimension

7. Proposition 4.8.3 - artinian rings and maximals

8. Example 4.8.1

9. Proposition 4.8.4

10. Proposition 4.8.5

11. Theorem 4.8.2 - equivalence between artinian and notherian

12. Remark 4.8.2 - local artinian

13. Proposition 4.8.6

14. Theorem 4.8.3 - Structure Theorem for Artin Rings

15. Proposition 4.8.7 - local artin

16. Example 4.8.2

9. Primary Decomposition

1. Remark 4.9.1 - radicals

2. Definition 4.9.1 - Primary

3. Proposition 4.9.1 - characterization of primarys

4. Example 4.9.1 - primary ideals

5. Proposition 4.9.2 - m-primary ideals

7. Definition 4.9.2 - Primary decomposition

8. Theorem 4.9.1 - Noether-Lasker primary decomposition

9. Corollary 4.9.1

10. Proposition 4.9.3

11. Proposition 4.9.4 - primary quotients

12. Definition 4.9.3 - associated primes

13. Theorem 4.9.2 (1st uniqueness theorem)

14. Example 4.9.2 - primary decomp not unique

15. Definition 4.9.4 - isolated and embedded primes

16. Proposition 4.9.5 - about isolated primes

17. Proposition 4.9.6 - associated primes of 0

18. Example 4.9.3 - nilradical and zero divisors.

19. Proposition 4.9.7 - primary ideals in localization

20. Proposition 4.9.8 - primary decomp under localization

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 2.1.1- 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

Part 3 - Isolated Singularities

1. Definition 2.3.1 - Isolated Singularity

2. Remark 2.3.1

3. Definition 2.3.2 - Removable Singularity

4. Definition 2.3.3 - Poles

5. Definition 2.3.4 - Essential Singularity

6. Examples 2.3.1

7. Definition 2.3.5 - Laurant Series

8. Theorem 2.3.1

9. Corollary 2.3.1

10. Corollary 2.3.2

11. Theorem 2.3.2

12. Remark 2.3.2 - Characterization of Isolated Singularities

13. Theorem 2.3.3 - Casorati-Weierstrauss

Part 4 - Residues and Residue Theorem

1. Definition 2.4.1 - Residue

2. Example 2.4.1 - Calculating Residues

3. Theorem 2.4.1 - Residue Theorem

4. *Applications of Residue Theorem

5. Building the Gamma function

6. Building the Riemann Zeta Function

Part 5 - Conformal maps

1. Definition 2.5.1 - Conformal linear Maps

3. Remark 2.5.1

4. Remark 2.5.2

5. Definition 2.5.2

6. Theorem 2.5.1

7. Remark 2.5.3

8. Examples 2.5.3

Part 6 - Möbius Transformations

1. Definition 2.6.1 - Möbius Transformation

2. Remark 2.6.1 - Some Notes on Möbius Transformations

3. Proposition 2.6.1

4. Proposition 2.6.1 - An operation on Möbius Transformations

5. Definition 2.6.2 - Riemann Sphere

6. Construction - Stereographic projection

7. Proposition 2.6.3

8. Theorem 2.6.1

9. Theorem 2.6.2

10. Lemma 2.6.1

11. Proposition 2.6.4

12. Definition 2.6.3 - Cross Ratio

13. Theorem 2.6.3

14. Theorem 2.6.4

15. A few Applications

16. Definition 2.6.4

17. Proposition 2.6.5

18. Theorem 2.6.5

19. Corollary 2.6.1

20. Theorem 2.6.6

21. Definition 2.6.5 - differentiation at infinity

22. Definition 2.6.6 - singularities at infinity

23. Examples 2.6.1

Part 7 - Schwarz Reflection principle

1. Theorem 2.7.1 - Theorem of Morera

2. Theorem 2.7.2 - Schwarz Reflection Principle

3. Theorem 2.7.3 - Generalized Schwarz Reflection Principle

4. Theorem 2.7.4 - Riemann Mapping Theorem

5. Theorem 2.7.5 - Caratheodary Theorem

6. Theorem 2.7.6 - Schwarz-Christoffel formula for a Map

7. Corollary 2.7.1 - Nice Schwarz-Christoffel formula

8. Application - Inverse Problem

9. Example of Schwarz Christoffel

Part 8 - More on Analytic Continuation

1. Definition 2.8.1 - disk chain

2. Proposition 2.8.1 - antiderivatives

4. Example 2.8.1 - natural log.

5. Definition 2.8.2 - Disk chain along a curve

6. Lemma 2.8.1 - Analytic Continuation along a curve

7. Definition 2.8.3 - Analytic Continuation along a curve

8. Corollary 2.8.1

9. Theorem 2.8.1 - Monodromy Theorem

10. Crash course on Homotopys of Paths

11. Definition 2.8.4 - Fundamental Group

12. Lemma 2.8.2 - Path connected components

13. Definition 2.8.5 - Simply Connected

14. Example 2.8.2 - Star Shaped Domain

Part 9 - Global Cauchy Theorem

1. Definition 2.9.1 - winding number

2. Remark 2.9.1

3. Lemma 2.9.1 - traffic rule

5. Definition 2.9.2 - Homologous to zero

6. Definition 2.9.3 - Homologous curves

7. Theorem 2.9.2 - Global Cauchy Formula

8. Theorem 2.9.3 - Global Cauchy Integral Theorem

9. Theorem 2.9.4 - Global Residue Theorem

10. Definition 2.9.4 - chains of closed curves

11. Example 2.9.1 - Two homologous chains.

12. Theorem 2.9.5 -

13. Definition 2.9.5 - Boundary

14. Definition 2.9.6 - Meromorphic

15. Theorem 2.9.6 - zeros - poles

16. Corollary 2.9.6 - Geometric interpretation

17. Theorem 2.9.7 - Theorem of Rouche

18. Corollary 2.9.7 - fundamental theorem of algebra

19. Corollary 2.9.8 - about rational functions

Part 10 - Convergence of holomorphic functions

1. Definition 2.10.1 - Local uniform convergence

2. Lemma 2.10.1

3. Theorem 2.10.1 - Weierstrauss

4. Theorem 2.10.2 - Theorem of Hurewicz

5. Example 2.10.1 - counter examples to previous theorem

6. Lemma 2.10.2 - pointwise dense convergence

7. Theorem 2.10.3 - Theorem of Montel

Part 11 - Decomposition of Meromorphich functions

1. Motivation 2.11.1 - partial fraction decomposition

2. Definition-Proposition 2.11.1 - Meromorphic functions with finitely many poles

3. Remark 2.11.1

4. Theorem 2.11.1 - Mittag-Leffler

5. Remark 2.11.2

Part 12 - Multiplicative Decomposition

1. Remark 2.12.1 - fundemental theorem of algebra.

2. Definition 2.12.1 - Convergence of infinite product

3. Theorem 2.12.1 - convergence with logarithm

4. Definition 2.12.2 - absolute convergence

5. Proposition 2.12.1

6. Corollary 2.12.1 - rearrange terms

7. Remark 2.12.2 - products are delicate

8. Theorem 2.12.2 - products of holomorphic functions

9. Remark 2.12.3 - formula for sin(z)

10. Theorem 2.12.3 (Weierstrauss)

11. Corollary 2.12.2 - Meromorphic representation

12. Remark 2.12.4 - Gamma function

Part 13 - Elliptic Functions

1. Definition 2.13.1 - periodic

2. Proposition 2.13.1 - observations about periods

3. Definition 2.13.2 - simply and doubly periodic

4. Definition 2.13.3 - Fundamental Domain

5. Theorem 2.13.1 - Louiville

6. Theorem 2.13.2 - residues of elliptic function

7. Corollary 2.13.1

8. Theorem 2.13.3 - degree of an elliptic function

9. Theorem 2.13.4 - sum of c-points

10. Construction 2.13.1 - Weierstrauss P-functions

11. Proposition 2.13.2 - p-functions are elliptic

12. Proposition 2.13.3 - decomposition of elliptic functions

13. Theorem 2.13.5 elliptic differential equation

14. Proposition 2.13.3 - Unproven about p-functions

15. Definition 2.13.4 - Cubic Curves

16. Theorem 2.13.6 - Addition Law for P-function

17. Definition 2.13.5 - Weierstrauss zeta function

18. Proposition 2.13.4 - properties of the weierstrauss zeta function

19. Application 2.13.1 - zeta function.

20. Example 2.13.1

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An isolated singularity $z_{0} \in U$ of $f : U \to C$ is called an essential singularity if 1;;it is not removable or a pole.

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6. Examples 2.3.1

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