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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.

Community
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Timothy Chow
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    +1, despite the fact that you're trying to systematise Feynman! :) – José Figueroa-O'Farrill Dec 31 '10 at 23:46
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    By the way Feynman mentions on p. 195 of the book how he lost his bet to Paul Olum that he “can do by other methods any integral anybody else needs contour integration to do.”

    $$ $$

    "So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration!"

     – Andrey Rekalo Jan 01 '11 at 01:51
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    Timothy, if your question is about "how to introduce an extra parameter in the right way", then there is no general way to do that. This is equivalent to finding, for a generic numerical identity $A=B$, some "non-trivial" (e.g., non-constant) functions $a(x)$ and $b(x)$ such that $A=a(x_0)$ and $B=b(x_0)$. It is exactly the matter of skills, intuition etc, to find out "the right way". So, would you accept the honest answer "no" to your question? – Wadim Zudilin Jan 01 '11 at 02:33
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    @Wadim: If you feel that my question is too vague and general, here is a more concrete version: Devise a technique of this type that successfully evaluates some class of definite integrals that is not evaluable by other known techniques (such as those implemented in existing computer algebra packages). In other words, I'm not asking for an algorithm for an undecidable problem, but just for something more systematic than a random flash of genius. – Timothy Chow Jan 01 '11 at 03:02
  • @Timothy, I am interested in exactly the opposite: "exotic" applications of the differentiating under the integral sign, like $$ \int_0^{1/\lambda}\frac{\sqrt{(1-\lambda p)(\lambda-p)}\log\dfrac{1-\lambda p}{p(\lambda-p)}}{(1-p^2)\sqrt p}\ d p =\pi\cdot{}_3F_2\biggl(\begin{array}{ } \tfrac12, , \tfrac12, , \tfrac12 \ \tfrac32, , 1 \end{array}\biggm| \frac1{\lambda^2}\biggr) $$ (see Lemma 3 in http://arxiv.org/abs/1012.3036 and several other evaluations there). The standard methods are in all these standard books, like Gradshteyn--Ryzhik... – Wadim Zudilin Jan 01 '11 at 03:43
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    @Wadim: Are our interests really opposed? Just as the Wilf-Zeilberger method rendered many formerly "exotic" hypergeometric identities routine and pushed forward the boundary of what we consider "exotic," so I am hoping to see some systematic method that changes our conception of what constitutes an "exotic" application of differentiating under the integral sign. Maybe your understanding of Gradshteyn-Ryzhik is deeper than mine, but it looks to me like a catalog of identities with references to scattered papers that use disparate techniques, rather than an exposition of a "standard method." – Timothy Chow Jan 01 '11 at 03:56
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    @Timothy, I have to agree with this point of view: I would love to have a uniform method, like WZ, to do the job. There are some to mention: the WZ theory (in its "original" paper) actually treats (multiple) integrals as well, but the paper by Gert and Doron is more about your requirements... There is a method of "brackets" by Victor Moll, but it is more related to the complex integration technique. And, yes, Gradshteyn--Ryzhik is much deeper than it could be thought of at first sight: many of the integrals are very challenging for doing yourself! – Wadim Zudilin Jan 01 '11 at 04:48
  • To do integrals like Feynman did, you have to practice a lot. You have to just do integral after integral by this method until you get a gut feel for it. You have to love evaluating integrals--grungy calculations involving calculus and algebra. I'm pretty sure Feynman started doing this before he went to MIT. So if you want to catch up, have at it. – drbobmeister Jan 01 '11 at 06:01
  • @Timothy Chow--can you give a reference, hopefully available online, for undecidability of evaluation of integrals? I've heard of such things before and would like to know more. – drbobmeister Jan 01 '11 at 06:03
  • Wouldn't Feynman be ashamed of being cited in a mathematical blog ? – Denis Serre Jan 01 '11 at 14:48
  • I doubt Richard "let's get organized" Feynman would disapprove of his systematisation. – optima Jan 01 '11 at 15:06
  • @Dennis Serre: I don't think so. – drbobmeister Jan 01 '11 at 18:31
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    @drbobmeister: What I'm hoping is that some MO reader has this gut feeling and can articulate it, thereby turning a black art into a science. Victor Moll for example loves evaluating integrals but when I asked him (a version of) this question back in June, he didn't have a ready answer. As for undecidable integration problems, I'd start with Richardson's paper, "Some undecidable problems involving elementary functions of a real variable," http://www.jstor.org/stable/2271358 – Timothy Chow Jan 01 '11 at 20:51
  • Nowhere near systematic, but I think this write-up tries to present/exposit the standard/most popular tricks in as organized/streamlined a way as possible: https://zackyzz.github.io/feynman.html (first found via https://www.reddit.com/r/math/comments/11xkdun/feynmans_trick/) – D.R. Apr 12 '23 at 06:56

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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".

José Figueroa-O'Farrill
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  • I thought $\int_{\mathbb{R}} p(x)e^{-\frac{1}{2}x^2}dx$ can be computed by integration by parts as well for any polynomial $p$ – Amr Feb 08 '22 at 12:24
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Even this does not address the original problem of differentiating under the integral sign, there is a partly successful way to calculate the the loop integrals associated to the Feynman diagrams. It is hard to describe the strategy, called by the authors method of brackets, therefore I refer to the original paper

I. Gonzalez, V.H. Moll, and A. Straub, The method of brackets. Part 2: examples and applications, Contemp. Math. 517 (2010), 157–171. doi:10.1090/conm/517/10139, arXiv:1004.2062 (pdf)

Let me cite the short section "Conclusions and future work" in the paper:

The method of brackets provides a very effective procedure to evaluate definite integrals over the interval $[0,\infty)$. The method is based on a heuristic list of rules on the bracket series associated to such integrals. In particular, a variety of examples that illustrate the power of this method has been provided. A rigorous validation of these rules as well as a systematic study of integrals from Feynman diagrams is in progress.

David Roberts
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Wadim Zudilin
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"Devise a technique of this type that successfully evaluates some class of definite integrals that is not evaluable by other known techniques (such as those implemented in existing computer algebra packages)."

Wait what about the Risch algorithm?

http://en.wikipedia.org/wiki/Risch_algorithm

optima
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    Note the adjective definite in "definite integral." But even for indefinite integrals, you can get undecidable problems by introducing non-elementary functions into the picture. – Timothy Chow Jan 01 '11 at 03:31
  • i see. definite vs. indefinite. never mind then. – optima Jan 01 '11 at 10:48
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    Note that "Risch algorithm" is not a proper algorithm. It requires to be able to determine if a given elementary expression is the exact zero, and there is no known algorithm for that. – Vladimir Reshetnikov Mar 23 '14 at 19:24