# The Calculus of Friendship

Taking it to the limit

If you like great math riddles, you will enjoy this book by Steven Strogatz about his 30 year correspondence with his high school math teacher. There are excellent discussions about Feynman’s differentiation under the integral, riddles involving spiraling bugs (Martin Gardner’s Mathematical Games), cylindrical gas tanks (NPR: Car Talk), the Monty Hall problem (“Let’s make a deal”), infinite series and steepest descent (epicycloid), among many others. Eric Kemer and I did some fun work on the spiraling bugs a about a year ago and have some other versions nice solutions to that one as well, if you’re interested. I am still working on my proof of the steepest descent problem but thought I would share the Feynman example since I hadn’t remembered reading about differentiating under the integral from Surely You’re Joking Mr. Feynman.

Feynman made good use of this strategy when evaluating integrals he did not know how to solve when in undergrad and graduate school (and even during the Manhattan Project). It is done easily. For example, given the interval that high school students learn for the exponential function. From here, we can think of doing something strange (to compute a new, related integral) and differentiate both sides with respect to a. We have generated a new function for which we can calculate the integral. If you do this n times, you can derive the gamma function (when a is 1). How cool is that!

And therefore:

$\int_0^\infty{t^ne^{-t}}=n!$