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content/Exam Preparation.md

@@ -5,19 +5,19 @@ The following list is meant to provide a starting point for the type of question
 ## Topology and Measure Theory
 Here are 20 relevant exam questions in the topology and measure theory part of the course:
 
-1. What distinguishes the real numbers from the rational ones? [[Question 1|Answer]]
-2. What is an equivalence relation? [[Question 2|Answer]]
+1. What distinguishes the real numbers from the rational ones? [[ACIT4330/Revision/Real Analysis/Question 1|Answer]]
+2. What is an equivalence relation? [[ACIT4330/Revision/Real Analysis/Question 2|Answer]]
 3. What is a topological space? Examples? [[Question 3|Answer]]
 4. What is the ball topology on a metric space? [[Question 4|Answer]]
 5. What is the topology on a Banach space? [[Question 5|Answer]] - TODO
 6. What is a compact set? [[Question 6|Answer]]
 7. State the Heine-Borel theorem. Proof? [[Question 7|Answer]]
-8. What is a continuous function?
-9. Why does a real valued continuous function obtain its maximum on a compact set?
-10. What is a net? Given an example of an upward filtered ordered set.
-11. What is the initial topology?
-12. What is the product topology?
-13. What is a measure? Easy examples?
+8. What is a continuous function? [[Question 8|Answer]]
+9. Why does a real valued continuous function obtain its maximum on a compact set? [[Question 9|Answer]]
+10. What is a net? Given an example of an upward filtered ordered set. [[Question 10|Answer]]
+11. What is the initial topology? [[Question 11|Answer]]
+12. What is the product topology? [[Question 12|Answer]] - TODO
+13. What is a measure? Easy examples? [[Question 13|Answer]] - TODO Example
 14. Define the Lebesgue integral of a extended non-negative measurable function.
 15. State Lebesgue's monotone convergence theorem.
 16. Define $L^p$-spaces, and point out their crucial property.
@@ -27,7 +27,7 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 19. What is a complex measure?
 20. State the Lebesgue-Radon-Nikodym theorem.
 ## Complex Analysis
-- Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$.
+- Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$. [[ACIT4330/Revision/Complex Analysis/Question 1|Answer]]
 - Holomorphic functions and their properties.
 - Complex exponential and logarithm.
 - Integration in the complex plane.

content/Revision/Complex Analysis/Question 1.md

@@ -0,0 +1,11 @@
+# Question
+Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$.
+# Answer
+## Similarities
+- Both have one $\mathbb{R}$ axis
+- They can express ODEs (even though in different ways)
+## Differences
+- Angles in $\mathbb{R}^2$ are usually expressed as $2\pi$, whereas $\mathbb{C}$ is in $\pi$
+	- In $\mathbb{C}$ quadrants 1 and 2 are $\pi$; quadrants 3 and 4 are $-\pi$
+- They cannot directly be translated to each other without losing data because imaginary numbers
+- You can multiply pairs in $\mathbb{R}^{2}$ such as $(x_{1},y_{1}) \times (x_{2}, y_{2}) = (x_{1}x_{2}, y_{1}y_{2})$ 

content/Revision/Complex Analysis/Question 2.md

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+# Question
+Holomorphic functions and their properties.
+# Answer