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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


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