The Exponential Function. I think this is the first time we’ve done it here. It won’t be the last. You could write a book about it, and someone probably has. Here’s the usual picture:
This graph isn’t as pretty as the usual, because I’m at home with my old copy of Mathematica 5 instead of the new version that’s much better at drawing smooth and professionally-colored graphs. Nonetheless, the essential details are made clear. The exponential function drops off to zero very rapidly as you go left, and increases rapidly as you go right.
You can define the exponential function in any of numerous equivalent ways. Generally the mathematician picks a way which is best suited to the task at hand and then derives the other definitions from it. In an elementary sense you might for instance define it as repeated multiplication as you would with 24 = 2 x 2 x 2 x 2 = 16, and then proceed to generalize this to fractional and irrational exponents before finally defining the number e = 2.71828…
And that would work. It wouldn’t be especially clear why you’d bother doing such a thing, but it works. But mathematicians are interested in interesting things and so they typically define the exponential function in a more lucid way. In introductory calculus it’s usually defined as the inverse of the logarithm, which is itself defined in terms of the integral of 1/x.
Here I want to point out the definition that’s usually the most insightful from a physics standpoint. The exponential function satisfies the differential equation
That’s the snazzy calculus way of saying that we’re looking for a function whose rate of growth is equal to the function’s value. It turns out that the exponential function satisfies that requirement. You might be able to intuit that from the graph: the higher it rises, the more rapidly it’s rising. Lots of functions do that, but only the exponential does it in such a way as to keep the function value and the rate of increase exactly equal at all points.
The exponential function is ubiquitous in physics. You’ll find it in quantum mechanics, in classical mechanics, in electrodynamics, nuclear, AMO, you name it. As a physicist you eat, sleep, and breathe this thing. Which isn’t a bad thing. There are much more complicated functions out there, and that the comparatively simple exponential works so well for so many things is nice.