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Prove that \begin{align} \sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2}\right) \, \, \, \, \forall \, z \in \mathbb{C} \end{align}

The hint I had it's to use the Fourier series, but I really don't see how. Any suggestions? Thanks in advance!

userr777
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  • So the above equality should also hold in $z=0$? – dan_fulea Mar 03 '19 at 20:50
  • Compare with the formula in https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem#The_Weierstrass_factorization_theorem ... – dan_fulea Mar 03 '19 at 20:52
  • @dan_fulea just edited, sorry... – userr777 Mar 03 '19 at 20:53
  • Usually, one uses the Weierstrass factorization theorem, which is one of the main themes in a complex analysis course, unavoidable. A proof using Fourier ideas may be found here... https://www.youtube.com/watch?v=tk_HfqDfHss or https://math.stackexchange.com/questions/674769/sinx-infinite-product-formula-how-did-euler-prove-it (+ many other google hits) . See also https://math.stackexchange.com/questions/786046/infinite-product-prod-limits-k-1-infty-left1-fracx2k2-pi2-right and https://math.stackexchange.com/questions/1204021/any-elementary-proof-for-eulers-product-formula-for-sine/1204275. – dan_fulea Mar 03 '19 at 21:55

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