Update src/1_complex_numbers.md

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).