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Re: Vieta infinite product formula

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83849] Re: Vieta infinite product formula
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 2 Dec 2007 04:12:22 -0500 (EST)
  • References: <20071201233246.712$Zx@newsreader.com>

On 2 Dec 2007, at 13:32, David W. Cantrell wrote:

>
> 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



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