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Re: Vieta infinite product formula

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83847] Re: Vieta infinite product formula
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Sun, 2 Dec 2007 04:11:18 -0500 (EST)
  • References: <fire06$riv$1@smc.vnet.net>

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.

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.

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


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