Basel problem

Below is a clip from the movie "The Eternal Vision of Leonhard Euler" which was written and directed by Jon Sauer in 2008. Thanks for John Lieb, who helped me getting to the media in May, 2012. The clip gives Euler's ingenious proof of the Basel problem. Euler wrote the sinc function sin(x)/x in two different ways: the first was the Taylor series:
   sin(x)/x = 1-x^2/3! + x^4/5! - ...  = (1-x22) (1-x2/(2 π)2) (1-x2/(3 π)2) ....
Comparing the terms for x2 on both sides and multiplying now gives the equation
  
       -1/6 = -1/π2 -1/(4  π)^2  - 1/(9  π)&2 - ....
which is equivalent to the Basel problem formula
        π2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...  .
This computation relies however on a theorem, which was rigorously proven only later. For entire functions, functions which have a Taylor series expansion which converges in the entire complex plane. According to the Weierstrass factorization theorem, one can write this as a product. More precisely, a theorem of Hadamard allows to establish that if the function grows only exponentially at infinity, and the roots appear in pairs z,-z, then one has a factorization into factors of the form form (1-a(k)/z) as well as a constant. In the case of the sin-series, one knows the roots, knows that sin(x)/x=[ e^(ix) + e^(-ix) ]/(2i x) grows only exponentially in |x| and that sinc(x)=sin(x)/x is equal to 1 at x=0. It is with Fourier theory (covered in 22b) that one can prove the Basel formula a bit more effectively.


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