This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating under the integral sign (a.k.a. the Leibniz integral rule):
When guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.
Some examples of this trick are provided on the Wikipedia page that I linked to above. What I am wondering is whether there is a systematic way to attack integrals via the introduction of an extra parameter and applying the Leibniz integral rule. By "systematic" I mean something that could be incorporated into the symbolic integration algorithms of a computer algebra package.
The closest thing I've found in the literature is the paper "The Method of Differentiating under the Integral Sign," by Gert Almkvist and Doron Zeilberger, J. Symbolic Computation 10 (1990), 571–591, which develops an algorithm for finding a differential equation satisfied by the integral
$$R(x) = \int_{-\infty}^\infty F(x,y)\ dy$$ when $F(x,y)$ is holonomic. However, typically the critical step in evaluating an integral "the Feynman way" is to figure out how to introduce an extra parameter in the right way, and the Almkvist–Zeilberger paper does not provide a systematic algorithm for this step.
The Wikipedia examples strike me as ad hoc, so the question I am posing to MO readers is, do you know of any heuristics for introducing extra parameters into integrals, that might form the starting point for a general algorithm? Anything that helps remove the black-magic or rabbit-out-of-a-hat aura of introducing extra parameters would be welcome.
Best Answer
This is not a complete answer, of course, but one example which I didn't see in the wiki page and which is somewhat paradigmatic, as it forms the basis on which the Feynman path integral is used in practice in order to compute correlation functions, is to add a linear term to a gaussian integral (known as a source in quantum-field-theoretical lingo): $$ I(\alpha):= \int_{\mathbb{R}} e^{-\frac12 x^2 + \alpha x} dx $$ and in this way, by differentiating with respect to $\alpha$, compute the expectation value of any polynomial (or even analytic) function of $x$. One could do this in principle to any distribution, but the charm of the gaussian is that the integral can be evaluated exactly: $$I(\alpha) = \sqrt{2\pi}\ e^{\frac12 \alpha^2}$$ This requires completing the square and using the translation invariance of the measure $dx$.
In fact, this is one of the fundamental assumptions in quantum field theory: namely, that the path integral "measure", however it is defined, had better be invariant under "translations".