Have you ever been very surprised that something has, or doesn't have, a closed form?

Question

I'm trying to develop my intuition about when something likely has, or does not have, a closed form expression. So I would like to ask:

Have you ever been very surprised that something has, or doesn't have, a closed form?

For example:

• I was once surprised that $\sum\limits_{k=1}^n \sin k$ has a closed form , and yet $\prod\limits_{k=3}^\infty \cos{\left(\frac{\pi}{k}\right)}$ does not. You would think that taking sines of integers would not lead to anything nice, whereas taking cosines of rational multiples of $\pi$ would.
• I once accidentally stumbled upon $\int_0^\pi \arcsin{\left(\frac{\sin{x}}{\sqrt{5/4-\cos{x}}}\right)}dx$ and was surprised that it has a closed form.

Among my recent questions, a chain of circles sometimes leads to a closed form, whereas a spiral of circles apparently does not, and I'm trying to get a sense of "why".

I hope my question is acceptable as a soft question. I think answers could be helpful and interesting.

Answer 1

Have you ever been very surprised that something has, or doesn't have, a closed form?

Yes, when first studying the Riemann Hypotesis I found out that you can express $\zeta(2n)$ in a closed form but there is no (known) formula for $\zeta(2n+1)$.

$$\zeta(2n)=\frac{2^{2n-1}\pi^{2n}|B_{2n}|}{(2n)!}$$

Answer 2

Consider the sequence $C_n = \frac{1}{n+1} {2n \choose n}$ of Catalan numbers. Now consider its Hankel determinants, the sequence of determinants of the Hankel matrices

$$H_n = \left[ \begin{array}{ccccc} C_0 & C_1 & C_2 & \cdots & C_{n-1} \\ C_1 & C_2 & C_3 & \cdots & C_n \\ C_2 & C_3 & C_4 & \cdots & C_{n+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ C_{n-1} & C_n & C_{n+1} & \cdots & C_{2n-2} \end{array} \right].$$

The surprising fact is that $\det H_n = 1$ for all $n$!

This example is totally mysterious if you don't know the relevant techniques (one of which is explained e.g. in this blog post; it can also be proven more directly using the Lindstrom-Gessel-Viennot lemma), but even after knowing the relevant techniques I still find it pretty mysterious. For many other mysterious determinant evaluations you can see Krattenthaler's Advanced Determinant Calculus.

Other Hankel determinants also have interesting "closed-form" evaluations, some of which are given in the linked blog post. For example if the Catalan numbers are replaced by the factorials then the corresponding sequence of Hankel determinants is $\prod_{i=1}^{n-1} i!^2$.

Answer 3

As others explained, it is very much up to how we define 'closed formula'.

To illustrate how you can cleverly cheat with the floor function and such, I present you Willan's formula (1964) to generate the n-th prime:

$$p_n=1+\sum_{i=1}^{2^n} \left\lfloor \left(\frac{n}{\sum_{j=1}^{i}\left\lfloor\left(\cos\frac{(j-i)!+1}{j}\pi\right)^2 \right\rfloor} \right)^{1/n}\right\rfloor $$ See https://youtu.be/j5s0h42GfvM for details.

Answer 4

As usual in Mathematics, we have to define the terms we use.

The thing is, it's not yet common practice to do that for the terms "in closed form" and "elementary function", and its definitions aren't yet widely known.

One way to define closed-form objects is generating them by repeated application of functions of a given class, e.g. by towers of fields.

see [Borwein/Crandell 2013]

Examples are the Liouvillian functions and the Elementary functions.

see What general kinds of closed-form problems are there?

Today, we have the works of Liouville, Ritt, Bronstein, Chow, Corless, Davenport, Khovanskii, Lin, Risch, Rosenlicht, Singer and others as well as symbolic summation.

Knowing this, it's no surprise that some mathematical objects can be represented in closed form and others not.

[Borwein/Crandell 2013] Borwein, J. M.; Crandall, R. E.: Closed Forms: What They Are and Why We Care". Notices Amer. Math. Soc. 60 (2013) (1) 50-65