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Old 2013-03-20, 12:57   #3
jinydu's Avatar
Dec 2003
Hopefully Near M48

175810 Posts

Thanks for the links. I already read Journey through Genius in high school; that's where I learned of this solution!

Going to have to think about the Weierstrass Factorization and it's application to this case; but as for the other concern, I still don't quite see how to prove the validity of (8) from Newton's identities. I mean, I'm comfortable with applying Newton's identities to get the coefficients of x^2, x^4, x^6 etc. if we had a finite product. The problem here is the product being infinite.

Seems to me it boils down to three steps:

8.1) The sequence of partial products (1-x^2/pi^2), (1-x^2/pi^2)(1-x^2/4pi^2), (1-x^2/pi^2)(1-x^2/4pi^2)(1-x^2/9pi^2), etc. converges to sinc(x). [Pointwise this is ok, but is that going to be enough?]

8.2) Use Newton's identities to find the x^2 (and x^4, x^6, etc. if I feel like it) coefficients of each of these partial products.

8.3) Argue that if {f_n} is a sequence of entire functions converging (is pointwise enough?) to an entire function f, then the x^2 (and higher) coefficients of the Taylor expansions of f_n converge to the x^2 coefficient of the Taylor expansion of f. [Come to think of it, I'm pretty sure I heard that in the real setting, even f_n -> f uniformly does not imply f_n' -> f'. In the complex setting, err... Not sure...]


Last fiddled with by jinydu on 2013-03-20 at 13:10
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