Re: Vieta infinite product formula

*To*: mathgroup at smc.vnet.net*Subject*: [mg83851] Re: Vieta infinite product formula*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sun, 2 Dec 2007 04:13:27 -0500 (EST)*References*: <20071201233246.712$Zx@newsreader.com> <04798D06-095D-4AA6-8004-57C0E811F1E4@mimuw.edu.pl>

On 2 Dec 2007, at 15:02, Andrzej Kozlowski wrote: > > On 2 Dec 2007, at 13:32, David W. Cantrell wrote: > >> [Message also posted to: comp.soft-sys.math.mathematica] >> >> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: >>> I just discovered, to my disappointment, that Mathematica does not >>> know the classic Vieta infinite product formula: >>> >>> Sin[x]/x == Product[Cos[x]/2^k, {k, 1, Infinity}] >>> >>> Shouldn't something be done about that? >> >> First, the result you have in mind is due to Euler actually. > > Well, actually this formula is rather trivial to derive so so I do > not think Euler would have minded my misattribution. Besides, one > should keep in mind the famous "Arnold principle" (http://pauli.uni-muenster.de/~munsteg/arnold.html > ) which states: > If a notion bears a personal name, then this name is not the name of > the discoverer. > So, in calling this Vieta's infinite product formula I simply > followed this well established principle. > > Actually, what is usually called Vieta's infinite product formula is > what you get when you substitute x=Pi/2 and express the right hand > side in tems of radicals. (I came across this while reading on a > train Mark Kac's - Statistical independence in probability theory, > analysis and number theory). >> >> >> Second, there is a typo. You intended to ask for >> >> Product[Cos[x/2^k], {k, 1, Infinity}] >> >> instead. But Mathematica (at least version 5.2) leaves that >> unevaluated. I >> would have hoped that version 6 would have given Sinc[x] as the >> result. > > Yes, indeed it does that. >> >> >> Third, and surely more surprising, Mathematica 5.2 leaves >> >> Product[Cos[x]/2^k, {k, 1, Infinity}] -- that is, _with_ the typo >> >> unevaluated! I had expected Mathematica to factor out the cosine >> and thus >> to treat the product as though it were >> >> In[14]:= Cos[x] Product[1/2^k, {k, 1, Infinity}] >> >> Out[14]= 0 >> >> That Mathematica failed to do that I consider to be more >> disappointing than >> its failure to recognize Euler's product as being Sinc[x]. >> >> David W. Cantrell > > > Not quite. You get a product of infinitely many Cos[x], so assuming > that x is real the result is indeed 0. But I suspect Mathemaitca is > not making this assumption. If you use any real number in place of > x, e.g 27, you will indeed get: > > Product[Cos[27]/2^k, {k, 1, Infinity}] > 0 > > > Andrzej Kozlowski > Of course even if x is complex the answer is still 0, since the numbers Abs[Cos[x]/2^k] eventually become all less than 1. But probably this already needs slightly too sophisticated mathematical reasoning for present day CAS. Andrzej Kozlowski