[HN Gopher] Ramanujan Factorial Approximation (2012) ___________________________________________________________________ Ramanujan Factorial Approximation (2012) Author : wglb Score : 47 points Date : 2022-09-08 03:32 UTC (19 hours ago) (HTM) web link (www.johndcook.com) (TXT) w3m dump (www.johndcook.com) | personjerry wrote: | How did he come up with it? | version_five wrote: | I don't know, other than he was very smart. But a potential way | that the article hints at is that the factorial is related to | the gamma function which I believe is the solution to a | differential equation... and so there may be some insight to be | had in looking at the relevant differential equation and trying | to approximate its solution. | | Edit after googling: the gamma function not the solution to a | differential equation, in fact there is a proof that it cant be | [*] but it's formulated as an integral, whose approximation | could lead to odd looking equations like the one in the article | | https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem | rq1 wrote: | Algebraic differential equation* | vouaobrasil wrote: | Ramanujan had a miraculous facility for just seeing expressions | without much formal derivation. He worked with the British | mathematician G.H. Hardy and it is clear from Hardy's | description and Ramanujan's biography that Ramanujan did not | have much insight into his own process or following clearly | defined steps. He must have thus had a very natural insight | into number theory that was mostly instinctual. | [deleted] | EpiMath wrote: | There was a paper a few years ago in the American Mathematical | Monthly ( sorry... can't remember the author, I think it was a | Russian mathematician ) that gave some interesting heuristics | for why this form is natural to consider, where the 1/30 comes | from ( and considerations of a couple of alternatives to the | "30" ), and the kind of intuition/thinking that Ramanujan may | have used. When you only see the final form of the equation | like this, it looks very mysterious and impossible that someone | could find it ( and to be fair, there are other results from | Ramanujan that are definitely in that class! ). Probably a | google search could find the paper, it was delightful to read. | mjcohen wrote: | I added this comment there: | | You can remove the overflow problems by modifying the code so | that it computes ln(x!) as | | lnfact = .5 _ln(math.pi)+x_ (ln(x)-1) | | lnfact += ln(((8 _x + 4)_ x + 1)*x + 1/30.)/6. ___________________________________________________________________ (page generated 2022-09-08 23:00 UTC)