From cfca3fc182c03cef3c3ab8dd7b5c858295d27b12 Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 20:24:52 +0000
Subject: [PATCH] Formatting problems sections
---
src/1_complex_numbers.md | 57 ++++++++++++++++++++++++----------------
1 file changed, 34 insertions(+), 23 deletions(-)
diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index ea01b4f..7bee6c8 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -288,26 +288,37 @@ function help in re-deriving trigonometric identities.
## 1.6. Problems
1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the
- complex plane the numbers $a+b$, $ab$, and $b/a$.
-
-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:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geometric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
-
-6. [:grinning:] Compute (a)
- $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
- and (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.
+ complex plane the numbers:
+ 1. $a+b$,
+ 2. $ab$,
+ 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.
--
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