Yesterday Nassim Taleb posted an interesting integral yesterday:

Here’s the integral:

Taleb frequently posts difficult problems and I have a thought about maybe 1 in 10 of them. This one caught my eye, though, because I’ve been reviewing basic complex analysis trying to understand more about recent work by Laura DeMarco’s and Kathryn Lindsey:

3-D Fractals Offer Clues to Complex Systems

Convex shapes and harmonic caps on arXiv.org

Anyway, I had 90 min this morning while my older son is at an archery class, so I thought it would be fun to write up the little exploration I had yesterday with this integral.

In Taleb’s integral, the substitution transform the integral (after a little algebra and ignoring some constants) to one in the form:

Other than the square term in the exponential (which will become a big deal shortly!) this integral looked a lot like one of the contour integral examples in my old complex analysis book:

Playing around a bit on Mathematica encouraged me that integrals in the form above did have closed form solutions:

Also encouraging was this paper I found online showing how to evaluate the usual Gaussian integral via countour integration (see example 9):

“The Gaussian Integral” by Keith Conrad

I was surprised that the countour integration solution was only recently (well, recently for math) discovered, but the really amazing coincidence is that the form of the integral used to evaluate the Gaussian integral via countour integration is nearly the same form as the integral we are trying to evaluate here. So, I dove in to the calculations but unfortunately didn’t have much luck finding a contour that worked.

Then this paper was posted on Math Stackexcange and on twitter:

The form of the integral had been studied before and, unfortunateley, the general solution was unknown. However, lots of interesting stuff was known including a recurrence relation that gives the values that Mathematica had found plus a few more. It is a surprise that you can know the values of these integrals when is an integer, but not in other situations. Another surprise was the odd form of periodicity discussed in the paper.

Along the way I also found this surprise on Mathematica, which probably has an easy analysis-related explanation but isn’t obvious to me geometrically:

So, despite not getting to a complete solution, this problem Taleb posted was a really fun one to study. It never ceases to amaze me how much fun math gets shared on Twitter.

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## Comments

Mike, indeed this integral is quite amazing. It is a particular case of the so-called Mordell integral, which was studied in 1933 by L.J.Mordell in this paper:

https://projecteuclid.org/euclid.acta/1485888015

It cannot be evaluated in closed form for general values of the x argument, although it turns out to be exactly known on a grid of points as you mentioned. Numerically it can be evaluated quite precisely by trapezoidal quadrature, and the error can be bounded by a neat contour integration argument, which is due to Alan Turing and others. I could send you the details of the proof if you are interested (the result appeared first in a paper by Crouch and Spiegelman).

Best,

Dan

Thanks – it was fun to explore, and the more I learn about it the less bad I feel for making so little progress!