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- Jan 31, 2012

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1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.

2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$.

3) Evaluate $ \displaystyle \prod_{k=2}^{\infty} \left(1- \frac{1}{k^{3}} \right)$.

2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$.

3) Evaluate $ \displaystyle \prod_{k=2}^{\infty} \left(1- \frac{1}{k^{3}} \right)$.

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