From a46353f39b6e42303eeda0b379c031c337a669ce Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 20:43:32 +0000
Subject: [PATCH] Update src/1_complex_numbers.md
---
src/1_complex_numbers.md | 21 ++++++++++-----------
1 file changed, 10 insertions(+), 11 deletions(-)
diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index e245814..60abfdc 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -287,30 +287,29 @@ 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:
+1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the complex plane the numbers:
1. $a+b$,
2. $ab$,
3. $b/a$.
-2. [:grinning:] Evaluate
+2. [:grinning:] Evaluate:
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>
+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 $b$.
+4. [:grinning:] *Quotients*</br>
+ 1. Find the real and imaginary part of $$ \frac{1+ {\rm i}}{2+3{\rm i}} \, .$$
+ 2. Evaluate for real $a$ and $b$:$$\left| \frac{a+b\rm i}{a-b\rm i} \right| \, .$$
-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?
+5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$.
+ 1. Visualize taking the inverse in the complex plane.
+ 2. What geometric operation does taking the inverse correspond to? </br>
(Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
-6. [:grinning:] </br>
+6. [:grinning:] *Differentation and integration* </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).
--
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