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ThudnBlunder
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 Sinc(x)   « on: May 17th, 2008, 10:47am » Quote Modify

Prove that sin(x)/x  =  cox(x/2)*cos(x/4)*cos(x/8)*cos(x/16) ...................
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pex
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 Re: Sinc(x)   « Reply #1 on: May 19th, 2008, 8:28am » Quote Modify

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

 hidden: 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).
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ThudnBlunder
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 Re: Sinc(x)   « Reply #2 on: May 19th, 2008, 10:52am » Quote Modify

Yep, that's it, pex.
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william wu

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 Re: Sinc(x)   « Reply #3 on: May 20th, 2008, 1:44pm » Quote Modify

There's a neat geometric interpretation of this formula in Eli Maor's book:

http://press.princeton.edu/books/maor/chapter_11.pdf
 « Last Edit: May 21st, 2008, 3:15pm by william wu » IP Logged

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