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Factoring a sine wave


Richard Baker
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If you look at the graph of sin (x) you see that all the zeroes are located at multiples of pi. So if you were to factor sin (x) as if it were a polynomial you would get m * x * (x-pi) * (x + pi) * ( x-2pi) * (x + 2pi)  * ( x - 3pi ) * (x + 3pi)  .....  (x - n pi) * ( x + n pi) as n approaches infinity.  Question:  what is m?  I figured this much from computer experiments, m is not a constant, it is a function of x and n.  m is less than 0 if n is odd,   m is greater than 0 if n is even.   

If you graph abs (x) and scaled it by the appropriate very large factor you'd see that m (x) looks like a bump centered at x =0 which morphs as n increases, it retains the bump figure. It is not of the form c * exp ( - k x2), although it looks like it could be. 

What is this mystery function m?   Please help me figure this out.  

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x * (x-pi) * (x + pi) * ( x-2pi) * (x + 2pi)  * ( x - 3pi ) * (x + 3pi)  .....  (x - n pi) * ( x + n pi)... seems a non-convergent sequence.

I wouldn't dare to touch this with a barge pole. ;)

But if I were in the mood of playing "being Euler", I would try to group (x-kpi)(x+kpi)=x²-k²pi² for integer k (not guaranteed to work, as reordering of infinite divergent sequences is dicy at best), take natural logs (to have sums instead of products) and work in the complex plane to see if I can "regularise" it somehow.

For example, it seems more sensible to me would be to introduce (guess at) some regularizing quotiens from the get-go.

...

Funny, I was thinking "this reminds me of something I've seen before" when I've come across this:

https://math.stackexchange.com/questions/674769/sinx-infinite-product-formula-how-did-euler-prove-it

But it's not just one m that you need. Every single factor needs a different m. Those would be the "regularising quotients" I suggested.

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