In my last post I briefly talked about Feynman’s favorite technique of integration – integration by differentiation. It’s pretty cool, and the thing is, it’s fairly intuitive. The integration technique I’m gonna show you next is nowhere near as intuitive but is nonetheless pretty damn awesome. I couldn’t find very many examples of it, so let’s jump right in.

Imagine you’re walking down a dark street one cold winter’s night when a man (or a woman – I won’t discriminate) in a mask points a gun to your head and says, “Solve the following integral, or else.”

Now I don’t know about you, but I probably wouldn’t be able to do this integral under such pressure (actually, I’m not sure if I’d be able to do it anyway, seeing as Wolfram|Alpha can’t do it, and parts doesn’t work). But if I were Feynman or Witten (I’ve heard he’s a decently smart guy) or someone, here’s something I might do:

First, let’s look at the much easier integral

.

Now let’s (yes, plural) have strokes of genius and multiply both sides of the above equation by and integrate from a to b:

Caveat: As a reader pointed out, it’d be nice to choose values of a and b such that the integral actually converges. On the left, since integration is commutative given that we consort only with sufficiently civilized functions, we can switch the order of integration and perform the inner integral. On the right, we can simply carry out the integration. We now get:

… HOLY CRAP.

I completely agree with you if you say this is not the most obvious way of approaching this integral. However, in hindsight, I can see why it works (powers and the logs that pop out when you differentiate)… but still.

Anyway, that’s all I have for today – I’m a bit tired after an especially fun/full-of-learning workday. If you have questions/ideas/statements-of-blown-minds, feel free to comment!

P.S. Anyone have any suggestions in terms of introductory (or slightly higher) books on quantum mechanics? I’m currently going through Griffiths, and for the most part it’s surprisingly clear and concise. It’s sometimes a bit muddy, however. For instance, in the section about the free particle, I felt like he didn’t do a spectacular job explaining that the eigenstates you get are inherently unphysical (and that you have to use an envelope/Fourier transform to get a general but also “localized” solution [or at least, that’s what I gathered from the stuff people said]). Apparently Columbia uses French and Taylor during freshman year.

P.P.S I just can’t get over the title of this post…