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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
6. Lemma 4.9.1
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
16. Example
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
2. Lemma 2.5.1
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
0. Review
1. Definition 2.8.1 - disk chain
2. Proposition 2.8.1 - antiderivatives
3. Warning
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
0. Remark
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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For any
c
≠
∞
, we have that:
∑
z
∈
P
,
f
(
z
)
=
c
z
−
∑
z
∈
P
,
f
(
z
)
=
∞
z
≡
0
m
o
d
(
L
)
PROOF:
Use a similar idea to the previous
theorem
The key part of this proof is that the branch of the logarithm is ==1;; not specified, but the argument is
2
π
periodic)
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