#realanalysis — Public Fediverse posts

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

I recently dumped my notes on modulus of convergence for hypergeometric functions on my website. I also had some thoughts on numerical accuracy. Not very valuable thoughts, but thoughts nonetheless.

For those who are wondering why anyone would care, many math libraries such as GSL, boost, and so on, suck. If you are trying to do intense calculations with any sort of decent accuracy, these libraries have dusty corners that fail. And they don't document where those corners are. Or they don't have routines for complex arguments. I write my own routines for these cases.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/numerical-accuracy-of-generalized-hypergeometric-series

A few years ago (mid 2023), I wrote up some research notes regarding the modulus of convergence of the generalized hypergeometric functions. This month I wrote up and posted a series of articles that are those notes, cleaned up a bit, and this article is the wrap-up of the series.

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence-for-generalized-hypergeometric-functions

We all get the feeling that, day by day, the world is converging towards disaster. But what tells us how fast? The Modulus of Convergence does!

#Math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/modulus-of-convergence

In mathematics, we say that a function is bounded if can restrict its image. Oddly, we never seem to 'bind' a function, though. I can find bounds on the the remainder of generalized hypergeometric functions, and I never used the word 'bind' either. Maybe 'bounding' refers to bunnies and deer?

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/remainder-of-the-conway-maxwell-poisson-distribution-normalization-factor

The generalized hypergeometric series are ubiquitous in the world of computing special functions, for certain amounts of ubiquity. Turns out the speed of convergence is related to the obscure Conway-Maxwell-Poisson distribution, which no one has ever heard of - pretty much the opposite of ubiquity.

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/bounding-the-remainder-of-generalized-hypergeometric-series

Accurately computing generalized hypergeometric functions is hard. How many terms do we need? Guess we need a general expression for the size of the terms in the series. Oh, wait, I've got one right here!

#math #RealAnalysis #NumericalComputing

https://www.skewray.com/articles/estimating-the-terms-of-generalized-hypergeometric-series

#realanalysis #complexanalysis #complexnumbers