We will now consider s-plane techniques. Not because we will use them, but more to understand some of the common electronics terminology.
We can enhance the usefulness of the transfer function 
by transforming to a complex frequency.
Define the complex variable 
such that
where 
is an inverse time constant.
Our exponential function now becomes
and we have a rich set of cases
So we can not only describe oscillatory behavior but transient responses as well.