I have the following function skatches of a complex exponential function $e^{\sigma + j\omega t}$: function
But the sketch does not make sense to me at all because if I worked it out right the real part of the function is periodic:
$e^{\sigma + j\omega t} = e^{\sigma} \cdot (\cos(\omega t) + j\sin(\omega t))$
So the real part would be...
$e^{\sigma} \cdot \cos(\omega t) \text{ with } \sigma = 0.1, \omega = 1\\\Rightarrow e^{0.1} \cdot \cos(t)$
...which seemingly is a periodic function which does not increase or damp anyway.
Where is my fallacy?
Glad for your help!
In the function of the picture a pair of brackets is missing:
$e^{\sigma + j\omega t}$ should be $e^{(\sigma + j\omega )t}$
Thats the magic!