@@ -287,27 +287,33 @@ 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
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
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:
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:] *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:] 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:] *Differentation and integration*</br>