I am given $\Psi(x)=A_0 e^{-kx^2} e^{i\alpha x}$
Here, $\large A_0=[\frac{1}{\pi \sigma_0^2}]^\frac{1}{4}$, $\large k=\frac{1}{2\sigma_0^2}$, and $\large \alpha=\frac{p_0}{\hbar}$
I want to find the modulus and the complex conjugate.
As long as $A_0$ is real, the complex conjugate looks like: $\Psi(x)=A_0 e^{-kx^2} e^{-i\alpha x}$
Now, I have to find the modulus. I know that is $z=re^{i\theta}$, then $|z|=r$. Going by that, $|\Psi|=A_0e^{-kx^2}$
(Is there any restriction on what $r$ can be? The reason I ask this question is because, in the back of my mind, I have $r=\sqrt{a^2+b^2}$ and $\theta=tan^{-1}\frac{b}{a}$. It seems rather odd that we can take $r$ to be whatever multiplies $e^{i\theta}$, given that $r$ and $\theta$ are both determined by the values of $a$ and $b$, where $z=a+ib$.)
My main question is: Given a complex function like the one above, what's the general method for finding its modulus? I'm a little confused, and the value of $|\Psi|$ mentioned above is taken from my notes. I'm not completely sure how it's arrived at.