Built on Facts

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.

Comments

#1 --bill
December 14, 2008

The exponential function satisfies the initial value problem dy/dx = y, y(0) = 1. The general solution to dy/dx = y is y = ke^x, where k is some constant. Existence and uniqueness is defined for solutions of differential equations together with an initial condition.
#2 Steve
December 14, 2008

Someone has written a book on it, at least on e, which is close enough: “e, the story of a number,” by Eli Maor is a very nice read which covers a lot more than just e.
#3 andre
December 14, 2008

“…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.”

Isn’t this true also for y(x) = 0?
#4 vvl
December 14, 2008

One could argue that y(x)=0 is a special case of y(x)=ke^x… namely k=0
#5 andre
December 14, 2008

Good point vvl. But I was hoping Matt would do y(x) = 0 as a Sunday Function with Mathematica 5 as it would be the most boring plot you could get.
#6 dWj
December 14, 2008

The positive reals form a Lie group under multiplication; I tend to think of the exponential as the map between that group and its algebra.
#7 Comrade PhysioProf
December 14, 2008

The positive reals form a Lie group under multiplication; I tend to think of the exponential as the map between that group and its algebra.

Me, too! HAHAHAHAHAH!
#8 Zeno
December 14, 2008

I, too, like Maor’s book about e, although one of my students had trouble when she went shopping for it at Barnes & Noble. When she asked at the assistance counter whether they carried “e: The Story of a Number,” the clerk smiled at her and said, “Oh, honey, e is a letter, not a number.” I guess you don’t need to pass Algebra 2 to work at B&N.

I also wanted to mention Walter Rudin’s Real Analysis, which starts off briskly in its first chapter by baldly stating, “This is the most important function in mathematics…” and proceeds to derive many of the essential properties of exp(x). A bracing exposition.
#9 Paul Murray
December 14, 2008

This formulation of the the exponential function makes it easy to figure out the polynomial expansion, even if you only have high-school math like me.

Let’s put a pin through it at f(0) = 1. So the polynomial expansion is
f(x) = … ?x^3 + ?x^2 + ?x + 1.

Now, if a polynomial has a derivative with a constant term of 1, then the first term must be 1.

f(x) = … ?x^3 + ?x^2 + x + 1.

If a polynomial has a derivative with a term 1x, then the x^2 term must be 1/2 (because the derivative of kx^2 is 2kx)

f(x) = … + ?x^3 + 1/2 x^2 + x + 1.
similarly,
f(x) = … + 1/6 x^3 + 1/2 x^2 + x + 1.

and so on. Of course, our pin was at an arbitrary location. Putting it elsewhere gives k e^x, as per comment #1.

This expansion reveals a number of cool things, including the fact that e^x grows faster than any finite polynomial.

Now – why is ln(x) related to the integral of 1/x ? Is it to do with ln(a)+ln(b) = ln(a*b)?
#10 CCPhysicist
December 14, 2008

I am terribly disappointed that you did not plot it in the complex plane. That is where e^z gets really interesting.
#11 Matt Springer
December 14, 2008

Indeed, CC. But that’s such a big topic that I’m saving it for its own Sunday Function later on.
#12 JK
December 15, 2008

For those who aren’t familiar with e^x, extending the x axis out to 7 or 8 would give a better sense of it’s most salient characteristic.
#13 kiwi
December 15, 2008

Paul,

Re your question about ln … yes

(area under 1/x from 1 to a) + (area under 1/x from 1 to b) = (area under 1/x from 1 to a*b)

This is easy to show using integration by substitution.
#14 kiwi
December 15, 2008

The power series for the exponential function is the COOLEST THING IN THE WORLD (sorry about the shouting but I just had to do it). If you use it to calculate, say, e^(-100) the terms grow to around 10^40 before finally decreasing and ultimately becoming tiny. After adding them all you get massive cancellation, the result being of the order 10^(-40). Do it at -1000 instead of -100 and those 40′s turn into 400′s. Do it for -10000 and … well you get the idea.
#15 David Derbes
March 1, 2009

When you do e^z (with z = x + iy), you might want to look at Paul J. Nahin’s “Dr. Euler’s Fabulous Formula”, and also his “i: An Imaginary Tale”. Eli Maor and Paul Nahin between them are doing very nice work following on the famous “History of Pi” by Petr Beckmann”, mathematical books on famous numbers and ideas for a reasonably general audience. Nahin’s later books have been more and more mathematical, which is fine by me.

If I had a ton of money, I’d put copies of most of Nahin’s and Maor’s books in high school libraries.
#16 JoeB
March 5, 2009

Physics Prof Al Bartlett (Colorado) wrote a series of six articles about the exponential function for The Physics Teacher magazine, in the 70′s, I think. His thesis was that this was perhaps the most important thing a high school or college student needed to learn, to function in the real world. He gave luncheon talks about it to Rotary Clubs and the like. The calculations can be done in one’s head, using the simplest arithmetic, using the fact that the doubling time (in years, say), multiplied by the percentaqe growth rate (per year), equals a constant, equal to 72 for real estate brokers and financial advisors (the rule of 72). My physics students, of course, learned that it is closer to 70, or 69.3, and why.
Growth of population or wealth or wages or prices are often nearly exponential over decades. It is fun for students to calculate when the population of the earth will reach ten persons per sq. ft., assuming continued growth at the current rate, or to project the cost of a postage stamp, a house, or going to the prom, backwards to their parents’ and grandparents’ days (they are amused), or forward to their granchildrens’ days (they are not so amused).