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   Author  Topic: Sinc(x)  (Read 4594 times)
ThudnBlunder
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Sinc(x)  
« on: May 17th, 2008, 10:47am »
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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 »
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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 »
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Yep, that's it, pex.   Smiley
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william wu
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Re: Sinc(x)  
« Reply #3 on: May 20th, 2008, 1:44pm »
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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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