Quartz sync: May 27, 2025, 11:42 AM

content/Exam Preparation.md

@@ -30,8 +30,8 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 - 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. [[Revision/Real Analysis/Question 3|Answer]]
-- Integration in the complex plane.
-- Cauchy's theorem and integral formula.
+- Integration in the complex plane. [[Revision/Real Analysis/Question 4|Answer]]
+- Cauchy's theorem and integral formula. [[Revision/Real Analysis/Question 5|Answer]]
 - Taylor and Laurent series.
 - Singularities and their classification.
 - Residues and the residue theorem.

content/Revision/Complex Analysis/Question 4.md

@@ -1,3 +1,11 @@
 # Question
 Integration in the complex plane.
 # Answer
+First consider the real integral $\int_{a}^{b} f(t) \, dt$ along the segment $[a,b]$.
+
+Then let $a,b \in \mathbb{R}$ and $f : [a,b] \to \mathbb{C}$ be continuous.
+Then we set $\int _{a}^{b} f(t) \, dt := \int _{a}^{b} \mathrm{Re} f(t) \, dt + i \int _{a}^{b} \mathrm{Im} f(t) \, dt$.
+
+## Example
+I don't know if it is needed but if you want one then
+[[Lecture 21 - Integration, Antiderivatives, Homotopies#]Lecture 21 - Integration, Antiderivatives, Homotopies#Integral along a Curve]

content/Revision/Complex Analysis/Question 5.md

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+# Question
+Cauchy's theorem and integral formula.
+# Answer
+## Cauchy's theorem
+Let $f : D \to \mathbb{C}$ be [[Holomorphic|holomorphic]].
+
+Suppose $\gamma_{1}$ and $\gamma_{2}$ are two ([[Smooth Parametrization|parametrized]]) closed curves in $D$ such that $\gamma_{1} \sim \gamma_{2}$. Then we have
+$$\int_{\gamma_{1}} f = \int_{\gamma_{2}} f$$
+## Cauchy's Integral Formula
+Let $C = \{ z \in \mathbb{C} : |{z - z_{0}}| = r \}$ be the circle radius $r$ centred at $z_{0} \in \mathbb{C}$, with positive orientation. Then
+$\oint_{C} \frac{1}{z-z_{0}} \, dz = 2\pi i$

content/Revision/Complex Analysis/Question 6.md

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+# Question
+Taylor and Laurent series.
+# Answer