From: "Dr. Michael Albert" Subject: Re: Infinite product for sin, cos? Date: Fri, 22 Jan 1999 23:46:30 -0500 Newsgroups: sci.math Keywords: Representations of functions as infinite products > Is there an infinite product representation of sin x or cos x? Thanks. > sin (pi * x ) / (pi * x) = x * (1-x)*(1+x) * (1-x^2)*(1+x^2) * (1-x^3)*(1+x^3) * . . . There are several proofs. A nice one which recently came to my attention is to apply Parseval's theorm[1] to the Fourier series for exp(i k * x) , then you integrate twice and exponentiate (or maybe differentiate twice and eponentiate--the details escape me and are left as an excercies :-)). Best wishes, Mike [1] That's the theorm relating the integral of the square of the function to the sum of the squares of the Fourier coefficients. ============================================================================== From: eclrh@sun.leeds.ac.uk (Robert Hill) Subject: Re: Infinite product for sin, cos? Date: Mon, 25 Jan 1999 13:48:28 +0000 (GMT) Newsgroups: sci.math In article , "Dr. Michael Albert" writes: > > Is there an infinite product representation of sin x or cos x? Thanks. > > > > sin (pi * x ) / (pi * x) > > = x * (1-x)*(1+x) * (1-x^2)*(1+x^2) * (1-x^3)*(1+x^3) * . . . I think that either the last x on the left side or the first x on the right side should be omitted. I think this formula is due to Euler (though he may not have proved it, certainly not in a way we would accept). It can be interpreted as saying that sin(x) is really just like a polynomial, except that it happens to be of infinite degree. We know that if we know all the complex zeros x_i of a polynomial (with multiplicities), we can reconstruct the polynomial as a scaling constant times the product of the factors x-x_i. The above formula says the corresponding thing for sin(x). I have an idea that there is a general theorem that if an entire function f(z) does not grow too fast as z -> infinity, then f(z) is an exponential times a possibly infinite product of linear factors corresponding to its zeros. In the case of sin(z) the exponential reduces to a constant. -- Robert Hill University Computing Service, Leeds University, England "Though all my wares be trash, the heart is true." - John Dowland, Fine Knacks for Ladies (1600) ============================================================================== From: spamless@Nil.nil Subject: Re: Infinite product for sin, cos? Date: 26 Jan 99 16:22:18 GMT Newsgroups: sci.math In sci.math Dr. Michael Albert wrote: > > Is there an infinite product representation of sin x or cos x? Thanks. > > > sin (pi * x ) / (pi * x) > > = x * (1-x)*(1+x) * (1-x^2)*(1+x^2) * (1-x^3)*(1+x^3) * . . . sin(pi*x)/(pi*x) = (1-x^2)(1-x^2/4)(1-x^2/9)...(1-x^2/n^2)... (PROD[(1-x^2/n^2):for n=1,2,3,...]) (the original cannot be true: it would have an zero of infinite multiplicity at x=1 and would have zeros at x=i, etc. sin(pi*x) has zeros at x=n for integral n.)