[HN Gopher] Ramanujan Factorial Approximation (2012)
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       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.
        
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       (page generated 2022-09-08 23:00 UTC)