Conditions on the real part of an exponential function If $a = \mu +i \omega$, what conditions are necessary to impose on $\mu$ and $\omega$ if $Re(e^{at})$ for $t>0$ is to be: a) exponential decreasing b) exponential increasing c) oscillating with constant amplitude d) oscillating with increasing amplitude e) oscillating with decreasing amplitude **This is what I think:** $Re(e^{at})$ = $e^{\mu t}$ so $\mu$ has to be negative for exponential decreasing and positive for exponential increasing while $\omega$ is zero. For it to be oscillating with constant amplitude $\mu$ has to be zero while $\omega$ can be any real number. What about d and e?

Your claim that "$Re(e^{at}) = e^{\mu t}$" is not correct.

If $a = \mu + i\omega$, then $e^{at} = e^{(\mu+i\omega)t} = e^{\mu t}e^{i\omega t} = e^{\mu t}(\cos(\omega t)+i\sin(\omega t))$.

So, $\text{Re}(e^{at}) = e^{\mu t}\cos(\omega t)$ not $e^{\mu t}$ as you have it.

To answer each of the parts, notice that the $\cos(\omega t)$ term is the oscillating term, and the $e^{\mu t}$ term controls the amplitude. Can you figure out the answers from this?