Quartz sync: May 27, 2025, 10:33 AM

content/Exam Preparation.md

@@ -7,9 +7,9 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 
 1. What distinguishes the real numbers from the rational ones? [[Revision/Real Analysis/Question 1|Answer]]
 2. What is an equivalence relation? [[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
+3. What is a topological space? Examples? [[Revision/Real Analysis/Question 3|Answer]]
+4. What is the ball topology on a metric space? [[Revision/Real Analysis/Question 4|Answer]]
+5. What is the topology on a Banach space? [[Question 5|Answer]]
 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? [[Question 8|Answer]]
@@ -17,7 +17,7 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 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
+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.
@@ -29,7 +29,7 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 ## Complex Analysis
 - Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$. [[Revision/Complex Analysis/Question 1|Answer]]
 - Holomorphic functions and their properties. [[Revision/Complex Analysis/Question 2|Answer]]
-- Complex exponential and logarithm.
+- Complex exponential and logarithm. [[Revision/Real Analysis/Question 3|Answer]]
 - Integration in the complex plane.
 - Cauchy's theorem and integral formula.
 - Taylor and Laurent series.

content/Revision/Complex Analysis/Question 3.md

@@ -0,0 +1,16 @@
+# Question
+Complex exponential and logarithm.
+# Answer
+## Complex Exponential
+The [[Complex Exponential|complex exponential]] $\exp : \mathbb{C} \to \mathbb{C}$ is defined by
+$$\exp(z) := e^{x}(\cos(y) + i \sin(y))$$
+which is also
+$$\exp(z) = e^{x} \times e^{iy}$$
+## Complex Logarithm
+In principle it is the inverse of $\exp(z)$
+
+But it is a function $\operatorname{Log} : \mathbb{C} \setminus \{ 0 \} \to \mathbb{C}$
+$$\operatorname{Log} z = \ln|z| + i \operatorname{Arg} z.$$
+>[!note] For $z = x \gt 0$
+>We get $\operatorname{Log} x = \ln x$.
+

content/Revision/Complex Analysis/Question 4.md

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+# Question
+Integration in the complex plane.
+# Answer

content/Revision/Real Analysis/Question 5.md

@@ -1,3 +1,6 @@
 # Question
 What is the topology on a Banach space?
 # Answer
+A [[Banach Space|Banach space]] is a [[Complete|complete]] [[Normed Vector Space|normed vector space]] $(X, \| {\cdotp} \|)$
+
+[[Normed Vector Space|Normed vector spaces]] are assumed to carry the [[Hausdorff]] [[Topology|topology]] unless stated otherwise.