Eventually, after the field was in terminal decline, set theorists went back and looked at the few open questions left, and discovered that they were all independent of the axioms of set theory. (This is the only thing I can think of that fits Raoul Ohio’s statement about the transfinite.) Set theory itself, while still active, is an incredibly unfashionable field, so unfashionable that people frequently use “core mathematics” on Math Overflow to mean “areas other than set theory”.

]]>http://www.damtp.cam.ac.uk/user/jono/uncertainty-nonlocality.html

http://www.wired.com/wiredscience/2010/11/entangled-uncertainty/

]]>Yes, it was great fun to read in that article’s historical appendix:

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Lambert had dreams [in the 1760s] of building a machine to do symbolic calculation … Thus his wishes anticipated those of Lady Ada Lovelace by roughly one century, and the actuality of symbolic computation systems by roughly two.

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This was news to me! ðŸ™‚

]]>Moreover, when these functions’ Padé approximants are extended to the complex *z*-plain, the observed Riemann structure is in reasonable accord with our emerging understanding of their analytic properties. Good!

I wonder, are there any other analytic functions that have largely or wholly originated as natural functions in complexity theory?

If *Shtetl Optimized* readers can identify more such “special functions of complexity theory”, please post about them! Because a review article titled *The Special Functions of Complexity Theory* would be too much fun to pass over! ðŸ™‚

I originally pursued these half-exponential calculations purely for fun … and to illustrate an essay—still in progress!—on theme of uniformity, naturality, and verification in STEM enterprises.

But now I am sufficiently encouraged to foresee also, that these functions might eventually—with much further work—be the subject of an article of similar scope and completeness to the classic 1996 article *On the Lambert W Function*, by Corless, Gonnet, Hare, Jeffrey, and Knuth. That would be great! ðŸ™‚

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… look for smooth analytic solutions of f_β(f_β(z)) = Î² exp(z) iff 0<β≤1/e … because it turns out that, viewed as an analytic function of the parameter β, the half-exponential function f_β(z) has analytic structure associated to the Lambert function W(-β), namely a cut (that by centuries-old convention is placed) on the positive real axis for β ∈ (1/e,∞).

For values of β along the Lambert cut, it turns out that analytic half-exponential functions can be constructed, but they exhibit “jumps” on the real axis (accompanied by the usual Gibbs phenomena) as is readily seen in the Padé approximants to f_β(z).

Thus (AFAICT) pretty much any reasonable approximation scheme will yield a smooth half-exponential function, iff the “exponential gain” β is be below the critical value β=1/e; the various pathologies and confusions that are reported in the literature thus are seen to be largely associated to choosing a too-large β.

We thus conclude—with reasonable confidence but not absolute certainty—that the half-exponential function f_β(z), considered as a doubly analytic function of z and β, is similarly well-defined and well-behaved to any other special function of mathematical physics.

Perhaps I will write a formal MathOverflow question that challenges the mathematical community to prove these claims in full rigor, with a view toward establishing f_β(z) as a respectable special function of informatic mathematics … that is, an function whose definition is simple and mathematically natural, and whose analytic structure is sufficiently well-characterized, and whose computation to any desired numerical accuracy is sufficiently efficient, as to be worthy of inclusion in standard compendia of special functions.

As Feynman once wrote:

“One thing is to prove it by equations; the other is to check it by calculations. I have mathematically proven to myself so many things that aren’t true. I’m lousy at proving things—I always make a mistake. […] So I always have to check with calculations; and I’m very poor at calculations—I always get the wrong answer. So it’s a lot of work […]”

Rigorously proving everything that concrete calculations are telling us about the analytic structure of half-exponential functions surely would be a lot of work … but it might be fun too. `Cuz people have been wondering about these functions for a long time! ðŸ™‚

]]>Namely, look for smooth analytic solutions of f(f(z)) = β exp(z) iff 0

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