Euler Formula for Hyperbolic Sine Function

Theorem

$\ds \sinh z$

$=$

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

$\ds $

$=$

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

$\ds \sin z$

$=$

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

$\ds \leadsto \ \ $

$\ds \map \sin {i z}$

$=$

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

$z \to i z$

$\ds \leadsto \ \ $

$\ds i \map \sinh z$

$=$

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

$\ds \leadsto \ \ $

$\ds \map \sinh z$

$=$

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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 38$: Infinite Products: $38.3$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 38$: Infinite Products: $38.3.$