@@ -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
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).