Back when I was in high-school I really enjoyed reading about Feynman’s entertaining numerical exploits. In particular, I remember integration challenges where he would use the Leibniz method of differentiating under the integral sign. I wasn’t taught this method at university but this detail resurfaced in my mind recently when I tackled a problem in hamiltonian dynamics where I had to differentiate under the integral sign. After using this method I decided to take a closer look at its mathematical justification.
For an integral of the form:
For all in some open interval, the derivative of this integral is expressible as
provided that and are both continuous over a region
we may deduce:
Now, if we define:
it’s clear that we may apply the Bounded convergence theorem where:
This is justified as the existence and continuity of combined with the compactness of closed intervals implies that is uniformly bounded.
Note 1: I plan to share the generalization of this result to higher dimensions in the near future.
Note 2: This blog post is almost identical to the Wikipedia entry with some modifications and clarifications which I find useful.