Exponentials

• The exponential function is typically used to describe the natural growth or decay of a system's state.

• An exponential function is defined as
  $$x(t) = e^{-t/\tau},$$
  where $e=2.7182...$, and $\tau$ is the characteristic time constant, the time it takes to decay by $1/e$.

  Figure 1: Exponentials with characteristic time constants, .1, .2, .3, .4, and .5

  

• Both exponential and sinusoidal functions are aspects of a slightly more complicated function.