Nothing lasts forever
Quick, what’s the integral from zero to infinity of ?
If you’re a good math student, you’ll tell me that the answer is undefined, since oscillates forever and so the integral doesn’t converge.
I, on the other hand, am not a good math student, so I am free to tell you that I know the answer.
The integral from zero to infinity of is .
Let me explain where this answer comes from, and why I’m so confident that it’s right. In doing so, perhaps I will demonstrate a little bit about the relationship that physics has with math.
First of all, as a physical-minded person I should interpret what it means to write . In my mind, the only meaningful interpretation of the question “what is ?” is something like “what is the net (integrated) effect of something that oscillates for a very long time?” The in the integral means that when someone asks “how long do you mean by ‘very long’?”, the correct answer is “as long as I want.”
Now I should answer the question, and I can do so as long as I hold to one belief: In the real world, nothing actually lasts forever.
That is, I don’t know how long, exactly, the in the integral will keep going, but I do know that it should die off eventually. So let me assume that the amplitude of the sine wave dies off very slowly (“How slowly?” “As slowly as I want.”). Then I can calculate the integral, get a perfectly well defined answer, and verify that my answer doesn’t depend on how slowly I killed off the sine wave.
For example, say I kill off the oscillations of the sine function exponentially, by replacing in the integral with , where is a very large number. Then I can calculate the integral, and check what happens when gets arbitrarily large. You can do this exercise for yourself (by hand if you’re diligent, using Wolfram Alpha if you’re lazy) and you’ll find that when the answer is very close to . (“How close?” “As close as you want.”).
The more precise mathematical statement goes like this:
(This is a mathematical trick that I use, in one form or another, all the time.)
So, to recap, what is the integral of from zero to infinity?
My math textbook, my math teacher, and all my math software says that the answer is undefined. But as long as you grant me that nothing actually lasts forever, I’ll tell you that the answer is .
1. It is not really my intention to bash on mathematics or math teachers. For example, I found that the most “hard core” math course that I took in college — real analysis — was thoroughly grounded in intuitive and physical thinking of the sort I am advocating here.
2. One way to think about the final answer, , is that the area under the curve above (red – blue) depends on how many red bumps and how many blue bumps you count. Every red bump contributes an area and every blue bump has area . So as you count them from left to right, your final tally for the area will go back and forth between and . The correct answer, , is the average of these two, which you might expect from any process that slowly washes out your counting procedure.
3. If you’re curious, . (This can of course be worked out using the same mathematical trick as above.)
4. If you’re really curious, . So the integral from zero to infinity of an oscillating wave can take any value from to , depending on its phase when it started. This result could be anticipated from the simple argument in Footnote 2.
5. Just to reassure you, there is nothing magical about the choice of an exponential cutoff . I usually use it because it’s easy to work with. But you’ll find that any slow damping of the oscillations will give the same result.
I suspect, in fact, that there is some nice theorem here. Like:
For any continuous function such that [UPDATE: and ], .
If I were a smarter person I could probably prove this theorem and generalize it to any oscillating function (with zero mean). Can any of my more mathematically inclined readers shed light on the subject? Maybe there are other necessary constraints on ?