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riddles >> putnam exam (pure math) >> Sinc(x)
(Message started by: ThudanBlunder on May 17th, 2008, 10:47am)

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Title: Sinc(x)
Post by ThudanBlunder on May 17th, 2008, 10:47am
Prove that sin(x)/x  =  cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ...................

Title: Re: Sinc(x)
Post by pex on May 19th, 2008, 8:28am

on 05/17/08 at 10:47:32, ThudanBlunder wrote:
 Prove that sin(x)/x  =  cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ...................

[hideb]
Repeatedly applying sin(2t) = 2 cos(t) sin(t), we find
sin(x) = 2 cos(x/2) sin(x/2)
sin(x) = 4 cos(x/2) cos(x/4) sin(x/4)
sin(x) = 8 cos(x/2) cos(x/4) cos(x/8) sin(x/8)
...
sin(x) = 2n sin(x/2n) * product[k=1..n] cos(x/2k)

Thus, for all positive integers n,
sin(x) / x = sin(x/2n) / (x/2n) * product[k=1..n] cos(x/2k)

Taking limits, we have
lim[n->inf] sin(x/2n) / (x/2n) = lim[t->0] sin(t) / t = 1

and therefore
sin(x) / x = product[k=1..inf] cos(x/2k).
[/hideb]

Title: Re: Sinc(x)
Post by ThudanBlunder on May 19th, 2008, 10:52am
Yep, that's it, pex.   :)

Title: Re: Sinc(x)
Post by william wu on May 20th, 2008, 1:44pm
There's a neat geometric interpretation of this formula in Eli Maor's book:

http://press.princeton.edu/books/maor/chapter_11.pdf