# Suggesting closed-form representations of mathematical constants by means of experimental mathematics?

Quite a few mathematical constants are known to arise in several branches of mathematics (more here). I have no doubt that they are useful to the understanding of some mathematical structures and add to the whole body of knowledge of mathematics as a whole. Sometimes, however, I find myself looking at these constants and I feel unsatisfied. I suspect that more constants can be expressed in more 'fundamental' constants like $\pi, e$ and $\gamma$.

Although Euler probably came close to be a living one, mathematicians of the past did not have computers. We can now use computers to determine te value of the afore mentioned constants with great precision. We can also try to conjecture the exact value of these constants by finding the value of some arithmetic combination of $\pi, e$ and $\gamma$ and the real numbers that corresponds to one of the 'unevaluated' constants with high precision. My question is: have experimental mathematicians 'found' the value(s) of some of the constants I described (by the the method I described or a similar one)?

Thanks,

Max

P.S. If someone knows of a reference to a paper/book that summarizes some results in this subield of a field, I would be grateful to him/her.

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I have seen many attempts to express the fine structure constant or mass(proton)/mass(electron) in terms of mathematical constants, then construct a physical theory that says it has to be that. None have been seen as convincing by the community, and many have fallen afoul of more accurate measurements of the physical constant they were modeling. – Ross Millikan Nov 8 '10 at 19:16
Bear in mind that almost all numbers are algebraically independent.... I see no reason that suggests certain constants can be written in terms of $\pi$. – anon Nov 8 '10 at 19:50
@ Qiaochu Yuan: I think the Inverse Symbolic Calculator is a useful tool, but even the smart lookup option only deals with a few combinations with $\pi$ and $e$ and some function, like ln(x). With a more advanced program, one can make millions of combinations with $pi$ , $e$ and the reals. Do you know of any of such project undertaken by experimental mathematicians? – Max Muller Nov 8 '10 at 19:54
Maybe you will find something interesting on this website (and in the books referred to there): experimentalmath.info – Hans Lundmark Nov 8 '10 at 21:32

I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from Experimentation in Mathematics by J.M. Borwein, D.H. Bailey and R. Girgensohn).

Example 1.

$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}-\epsilon,$$ where $0<\epsilon<10^{-41}.$

Example 2.

$$\sum\limits_{k=1}^{\infty}e^{-(k/10)^2}\approx5\sqrt\pi-\frac{1}{2}=8.362269254527580...$$

Well, they agree through 427 (four hundred twenty seven) digits yet they are not equal.

A moral. Make sure you understand the context and use your inverse symbolic calculator with caution.

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Example 2 is one of those theta series with exceedingly fast convergence. The reason why some PDEs are easy to numerically approximate is due to this speed. Here BTW is the article that preceded the Experimental Mathematics book by Borwein et al. – J. M. Nov 9 '10 at 0:37
@J. M.: That's right. Thanks for the reference. – Andrey Rekalo Nov 9 '10 at 0:44
@ Andrey Rekalo: Thanks a lot, this is what I what I was looking for. – Max Muller Nov 9 '10 at 20:12
@Max Muller: You're welcome. – Andrey Rekalo Nov 9 '10 at 20:42

Have you tried the Inverse Symbolic Calculator?

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@ Yuval Filmus: no, seeing this website is a first for me. Thanks! It can help to conjecture the value of some constants, although it would take a lot time to try some examples, whereas a good computer program probably can check a lot of values within a shorter time. I think I should point out that I'm more looking for a paper or another reference that presents the method by which closed-form representations of these constants and which "results" (conjectures) it has found already. Thanks for the suggestion, anyway! – Max Muller Nov 8 '10 at 19:25
@ Yuval and Qiaochu: I'm trying it now but I'm not able to insert any of the constants I want. I guess one has to punch in the 'mathematica-symbol', but what's the mathematica-symbol of Catalan's constant, for example? And how do I find the mathematica-symbol of other constants? – Max Muller Nov 8 '10 at 19:28
@Max Muller: that's not how it works. You input decimal expansions and the ISC tries to guess what the constant is. – Qiaochu Yuan Nov 8 '10 at 19:30
@ Qiaochu Yuan: ok... but it also says one can enter a maple expression (I meant that one, not a mathematica-symbol), how do you do that? – Max Muller Nov 8 '10 at 19:32
For example, is Catalan's Constant C in maple? And what about $\zeta(3)$? – Max Muller Nov 8 '10 at 19:35

The rational constant for an Apery like series for $$\zeta(4)$$ was found experimentally, using continued fractions.

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@ muad: interesting, I was going to post a question that somewhat relates to this. Thanks, muad. – Max Muller Nov 8 '10 at 19:39
"A numerical test (suggested by Cohen) implies that $\zeta (4)=\frac{\pi ^{2}}{90}=\frac{36}{17}\sum_{n=1}^{\infty }\frac{1}{n^{4}\binom{2n}{n}}$ (...) Apparently such expressions can be generated virtually at will on using appropriate series accelerator identities. Most startling of all though should be the fact that Apery's proof has no aspect that would not have been accessible to a mathematician of 200 years ago." ega-math.narod.ru/Apery1.htm – Américo Tavares Nov 19 '10 at 16:07

Gauss discovered the AGM/elliptic integral connection through the value $$\frac{\omega}{\pi}$$ by experimental mathematics.

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Not many people know that WA is also able of giving you all kinds of possible closed forms concerning fundamental constants, e.g.:

http://www.wolframalpha.com/input/?i=2.9299372410244

You can also click "more"...

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