Euler Formula for Sine Function

Theorem

$\ds \sin x$

$=$

$\ds x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }$

$\ds $

$=$

$\ds x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm$

for all $x \in \R$.

$\ds \sin z$

$=$

$\ds z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2 \pi^2} }$

$\ds $

$=$

$\ds z \paren {1 - \dfrac {z^2} {\pi^2} } \paren {1 - \dfrac {z^2} {4 \pi^2} } \paren {1 - \dfrac {z^2} {9 \pi^2} } \dotsm$

for all $z \in \C$.

This entry was named for Leonhard Paul Euler.

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

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 38$: Infinite Products: $38.1$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 38$: Infinite Products: $38.1.$