Can you help me to prove the following formula $$ \frac{\sin (\pi x )}{\pi x} = \prod\limits_{n \geq 1} \left(1 - \frac{x^2}{n^2}\right) ? $$
Thank you so much!
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Kenta S
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ToThichToan
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8That's in every decent textbook. MSE isn't a replacement, it's about when you don't get what the textbook says, and ask about the specific problem. – Oct 01 '20 at 17:43
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@ProfessorVector Thank you so much! In fact, it is a formula in my course, but I can not find a reference of its proof. Can you help me to share such one? – ToThichToan Oct 01 '20 at 17:54
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1@mathJuan might be of some help , and this as well – Anindya Prithvi Oct 01 '20 at 17:59
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Another related thing I met recently. – metamorphy Oct 02 '20 at 10:26
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Quick proof (not giving the details as you can find them in any textbook talking about the subject) : $$ \forall z\in\mathbb{C},\left(1+\frac{iz}{2p+1}\right)^{2p+1}-\left(1-\frac{iz}{2p+1}\right)^{2p+1}=2iz\prod_{k=1}^p\left(1-\frac{z^2}{(2p+1)^2\tan^2\left(\frac{k\pi}{2p+1}\right)}\right) $$ Taking the limit as $p\rightarrow +\infty$ gives $$ e^{iz}-e^{-iz}=2iz\prod_{k\geqslant 1}\left(1-\frac{z^2}{k^2\pi^2}\right) $$ and thus $\displaystyle \frac{\sin(z)}{z}=\prod_{k\geqslant 1}\left(1-\frac{z^2}{k^2\pi^2}\right)$.
Tuvasbien
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