# A math exploration via an interesting integral from Nassim Taleb

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 $u = e^z$ 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 $k$ 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.

# Buckets of Fish and Defeating Hydras

[sorry for a quick and unedited write up – wanted to get this out before we headed out for the day . . . .]

Saw a really neat post from Joel David Hamkins thanks to this tweet from Patrick Honner:

Buckets of Fish by Joel David Hamkins

We started today’s project by talking through the game. The question of whether the game ends in a finite number of steps was a little harder for them to understand than I think – but they got it eventually.

The confusion seemed to be the difference between:

(i) Here’s a path through the game than does end in a finite number of steps, and

(ii) there is no path that has an infinite number of steps.

After the short introduction to “Buckets of Fish” we watched Kelsey Houston-Edwards’s video on “Killing the Mathematical Hydra”. We’d watched this video when it came out a few weeks ago but didn’t use it for a project. I was happy that the boys remembered seeing it, though.

After viewing Houston-Edwards’s video we returned to whiteboard to talk about Hydras. My younger son did a nice job summarizing the hydra game, which I think is a testament to how good Houston-Edwards’s videos are explaining mathematical ideas to the public.

Next up – how are the hydras related to the fish?

Finally, to wrap up the project I thought it would be fun to study the case with 2 buckets more carefully. The motivation for this last section was a combination of the induction argument in Hamkins’s blog post and the introduction to induction proofs in Houston-Edwards’s latest video. For our project on that video see here:

Kelsey Houston-Edwards’s “Proof” video is incredible!

Anyway, I think the buckets of fish game makes for a nice introduction to mathematical induction for kids.

Definitely a fun project – sorry the write up is so rushed, but I wanted to get this out the door before we had to run out the door ourselves today 🙂