Identities in Higher Calculus and Special Functions | Hacker News Reader

Identities in Higher Calculus and Special Functions

fractional-calculus.com

64 pointsmbustamanter10 comments

I absoultely admire creativity and skills it takes to pursue developement of novel branch/framework of mathematics like this Super Calculus. I think that such original thinking at the fundamentals of mathematical formulations that underlying physical theories is required to overcome at least some difficulties modern physics struggles with. The problem is that it's nearly impossible to tell which one (even combination) is the right one.

The process to create one is so long and difficult, that there often isnt enough time to pursuit and find applications in which these novel mathematics coud prove to be superior over the existing ones. It creates a chicken and egg problem, where there are no arguments strong enough for the physics practicioners to switch these new formulations (as it takes a lot of practice and time with no guarantee of being any better than classical mathematics) and, on the other hand, the creators of these theories dont have enough time/audience/manpower/practice in practical calculations to go as far as where the problems of modern physics lie.

I've been thinking about this problem for some time and I think the right kind of design of the symbolic manipulation and calculation software could be of great help. The design of such software is certainly not an easy task, but hopefully somewhere along the road I, and hopefully others, will find some time and creativity to get started with it.

I second this thinking. You may be interested in reading about Geometric Algebra[1] and Infinitesimal Calculus[2], as two other alternative frameworks for the symbolic tooling used in the physical sciences. Together with symbolic manipulation and calculation software, I also think alternative, simpler symbolic frameworks may help solve what I call the "receding shoulders climbing problem". There's a phrase attributed to Newton "If I have seen further it is by standing on the shoulders of Giants". The problem today is that the giants became so big, one can spend many years studying and still die before getting anywhere near their shoulders.

[1]. https://arxiv.org/abs/1205.5935 [2]. https://en.wikipedia.org/wiki/Nonstandard_calculus

Yes! Those are exactly some of the alternative frameworks I had in mind. Also things like using differential algebra instead of classical epsilon-delta analysis and using Liouville's theorem to calculate integrals gives similar vibes.

Large parts of constructive mathematics seem to also be aligned well with physics and engineering (using only things that can be explicitly constructed) and hopefully could lead to interesting mathematical physics results. Same with, for example, quaternion analysis, which is very rarely used (contrarily to complex analysis), due to difficulties in operating quaternion valued functions, or computable analysis with its surprising result about differentiation of real an complex functions. I get similar feelings about non-classical logics. Somewhat more fringe examples that come to my mind, but are also interesting are for example non-Diophantine arithmetics or Holors theory (tensor generalisations).

Thanks for the "receding shoulders climbing problem" it’s a very nice way to frame the problem. I agree that there is a lot to be done to make shoulder-climbing process faster, easier and more widely available. It should also be possible, to make it more easy by for example storing not only results, but also derivations of many mathematical relations and computations. E.g. nowadays, there is no reason to put only results of integration in the integration tables, but whole derivations should be available as supplementary materials (preferably even in some form of symbolic computing code; this is somewhat realised with RUBI – rule based integration package). Size of the paper book is no longer the problem, so such “interactive derivations catalogues” should be extended to as many possible branches of mathematics, physics and engineering as possible.

Your comment reminds me of an old math professor of mine that told an anecdote about some famous mathematician that used to have a very promising student. However, after a while the student left and a colleague to the famous mathematician asked about the student. The mathematician said: "He lacked creativity, so he left mathematics and became a writer."

That “some famous mathematician” was David Hilbert. When he heard that one of his students had dropped out to study poetry, he said, “Good, he did not have enough imagination to become a mathematician.”

What I found most interesting was the sum convergence methods for multiple series.[1] Apparently with simple modifications you can improve the convergence rate of a multiple series sum by many orders of magnitude, making many multiple sums tractable which weren't before, using relatively simple identities that involve modifying both the summand and the limits of the indices.

Off topic: I like to think that combinatorics and statistics have a good-hearted sibling rivalry, both trying to explain what the other cannot but stealing from each others' crib notes all the time. That they meet in the middle with something as expressive as a calculus comes as little surprise.

[1] https://fractional-calculus.com/convergence_acceleration_mul...

Probably the weirdest math I’ve ever encountered is the Frobenius method and more generally power series methods of solving differential equations. I remembered encountering Taylor and Maclaurin series in Calc I, and thinking “oh, that’s a cute trick.”

A decade later, I’m casually working my through an introductory QM textbook, and realize that somebody had actually put the stuff to good use. It felt like power series methods are literally just tricking the hard differential equations around single electron 3D atoms into giving up their secrets.

> encountering Taylor and Maclaurin series in Calc I, and thinking “oh, that’s a cute trick.” ... somebody had actually put the stuff to good use

Aren't Taylor series expansions how calculators compute sine and cosine values?

Not usually. https://en.wikipedia.org/wiki/CORDIC is a bit more common, or if they are using an approximation, some form of min-max rational function approximation.

Oh I mean you could close your eyes and throw a dart at the map of mathematics and find any clever application thanks to <<insert-name-here>>’s ingenuity. I’m just saying that I had, as of yet, not known about or encountered a significant use of Maclaurin series.

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