A math exploration via an interesting integral from Nassim Taleb

Yesterday Nassim Taleb posted an interesting integral yesterday:

For those into integration, a difficult integral. https://t.co/G1mHgZE3Kc

— Nassim Nicholas Taleb (@nntaleb) March 10, 2017

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:

1/2 looks like a potential candidate for contour integration – for ex – from my old complex analysis book. Trouble is the square. pic.twitter.com/iVUeVrxCPo

— Mike Lawler (@mikeandallie) March 10, 2017

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:

logistic-normal integral. https://t.co/HCHXzUVsYM

— Arthur B. 🚀 (@ArthurB) March 10, 2017

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.