Euler Formula for Sine Function

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Theorem

Real Numbers

$\displaystyle \sin x = x \prod_{n \mathop = 1}^\infty \left({1 - \frac {x^2} {n^2 \pi^2}}\right)$

for all $x \in \R$.


Complex Numbers

$\displaystyle \sin z = z \prod_{n \mathop = 1}^\infty \left({1 - \frac {z^2} {n^2 \pi^2}}\right)$

for all $z \in \C$.


Source of Name

This entry was named for Leonhard Paul Euler.


Historical Note

The Euler Formula for Sine Function was not put on an adequately rigorous footing until Karl Weierstrass provided a satisfactory proof for Weierstrass Factor Theorem.