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===Infinite product expansion===
The following infinite product for the sine is due to [[Leonhard Euler]], and is of great importance in complex analysis:<ref>Whittaker and Watson, p 137</ref>
:<math>\sin z = z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \pi^2}\right), \quad z\in\mathbb C.</math>
For the proof of this expansion, see [[Sine#Partial fraction and product expansions of complex sine|Sine]]. From this, it can be deduced that
:<math>\cos z = \prod_{n=1}^\infty \left(1-\frac{z^2}{(n-1/2)^2 \pi^2}\right), \quad z\in\mathbb C.</math>
The first of these is obtained from the partial fraction decomposition of <math>\cot z</math>, which is the logarithmic derivative of <math>\sin z</math>.<ref>Ahlfors, p 197</ref>
=== Euler's formula and the exponential function ===
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