Modulus of the complex exponential
= Modulus of the complex exponential
{title2=$|e^z|=e^{\operatorname{Re}z}$}
For real $x,y$, the <complex exponential function> has $e^{x+iy}=e^x(\cos y+i\sin y)$. Since the trigonometric factor lies on the <unit circle>,
$$
|e^z|=e^{\operatorname{Re}z}.
$$
Thus the <real part> of the exponent controls its <modulus>, while the <imaginary part> controls its direction on the complex plane.