From a1e126ecba5c1da016a9d6bf6589b18480504c95 Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 20:38:14 +0000
Subject: [PATCH] Update src/1_complex_numbers.md
---
src/1_complex_numbers.md | 53 +++++++++++++++++++---------------------
1 file changed, 25 insertions(+), 28 deletions(-)
diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index 474a189..e245814 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -293,31 +293,28 @@ function help in re-deriving trigonometric identities.
3. $b/a$.
2. [:grinning:] Evaluate
- (a) $\rm i^{1/4}$,
- (b) $\left(1+\rm i \sqrt{3}\right)^{1/2}$,
- (c) $\exp(2\rm i^3)$.
-
-3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ (
- i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
-
-4. [:grinning:]
- (a) Find the real and imaginary part of
- $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$
- (b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
- for real $a$ and $b$.
-
-5. [:sweat:]
- 1. For any given complex number $z$, we can take the inverse $\frac{1}{z}$.
- 2. Visualize taking the inverse in the complex plane.
- 3. What geometric operation does taking the inverse correspond to?
- (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
-
-6. [:grinning:]
- (a) Compute $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
- (b) calculate the real part of $$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$
- ($k$, $x$, $\omega$, $t$ and $\gamma$ are real; $\gamma$ is positive).
-
-7. [:smirk:]
- Compute
- $$\int_{0}^{\pi}\cos(x)\sin(2x)dx$$
- by making use of the Euler identity.
+ 1. $\rm i^{1/4}$,
+ 2. $\left(1+\rm i \sqrt{3}\right)^{1/2}$,
+ 3. $\exp(2\rm i^3)$.
+
+3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ </br>
+ (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
+
+4. [:grinning:] </br>
+ 1. Find the real and imaginary part of $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$
+ 2. Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$ for real $a$ and $b$.
+
+5. [:sweat:] </br>
+ 1. For any given complex number $z$, we can take the inverse $\frac{1}{z}$.
+ 2. Visualize taking the inverse in the complex plane.
+ 3. What geometric operation does taking the inverse correspond to?
+ (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
+
+6. [:grinning:] </br>
+ 1. Compute $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
+ 2. Calculate the real part of $$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$
+ ($k$, $x$, $\omega$, $t$ and $\gamma$ are real; $\gamma$ is positive).
+
+7. [:smirk:] Compute by making use of the Euler identity.
+ $$\int_{0}^{\pi}\cos(x)\sin(2x)dx$$
+
--
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