This is maybe math $101$ question:
Let $z_1=1+i$.
I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$
But now if I take a look at
$z_2=-1-i$,
I know that $r=\sqrt 2$ and $\theta=\arctan(-1/-1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}}.$$
But $z_2$ should be equal to $$\color{red}{\sqrt 2 e^{5i\pi/4}} .$$
Why in $z_2$ should I add $\pi$ in the power of the exponent?
On
It is a matter of definition. Usually they define an angle called the principal angle in the interval $[-\pi,\pi)$. But you can choose your interval to be $[0,2\pi)$ too. Then you should look up for the unique angle in those intervals such that
$$\begin{cases} \cos \theta = \frac{x}{\sqrt{x^2+y^2}} \\ \sin \theta = \frac{y}{\sqrt{x^2+y^2}} \end{cases}$$
where
$$z=x+iy$$
Here's one way to view this: The arctangent function $\arctan m$ returns the unique value $\alpha \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\tan \alpha = m$, that is, the signed angle $\alpha$ in that range between the $x$-axis and a line with slope $m$.
Now, we can write any point $z = x + i y \in \Bbb C$ not on the $y$-axis as $\rho e^{i \theta}$ for a unique $r \in \Bbb R$ (not necessarily positive) and $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$: Here, $\theta$ is the signed angle determined by the slope of the line through $0$ and $z$ and $\rho$ is the signed distance from $0$ to $z$, where we take $r$ positive if $x > 0$ and $r$ negative if $x < 0$. The slope of the line is $\frac{y}{x}$ and so $\theta = \arctan \frac{y}{x}$.
For example, for $z_2 = -1 - i$, we have $\rho = -\sqrt{2}$ (since $x < 0$) and $\theta = \arctan \frac{(-1)}{(-1)} = \frac{\pi}{4}$ and so may write $$z_2 = -\sqrt{2} e^{i \pi / 4}.$$ On the other hand, for many purposes we prefer to have the radial component positive. We can remember this for our example (and similarly any case where $x < 0$) by writing $-1 = e^{\pi i}$, and substituting gives $$z_2 = (e^{\pi i}) \sqrt{2} e^{i \pi / 4} = \sqrt{2} e^{5 \pi i / 4}$$ as desired.
We can summarize this as writing $z = x + i y \in \Bbb C$ (with $x \neq 0$) as $$r = e^{i \theta},$$ where $r = |z|$ and $$ \color{#bf0000}{\boxed{\theta = \left\{\begin{array}{rl}\arctan \frac{y}{x}, & x > 0\\ \pi + \arctan \frac{y}{x}, & x < 0\end{array}\right.}} $$ In short, one "must add $\pi$" when $\Re z = x < 0$.
By construction, we have $\theta \in \left(-\frac{\pi}{2}, \frac{3 \pi}{2}\right),$ but notice that, since $e^{2 \pi i k} = 1$ for all $k \in \Bbb Z$, we can write $r e^{i \theta}$ as $r e^{i (\theta + 2 \pi i k)}$ for any $k$, so there are many choices of functions that map complex numbers to corresponding angles. These are call branches of the argument.
See the closely related function
atan2()used in many programming languages.