Euler Formula for Hyperbolic Cosine Function

Theorem

$\ds \cosh z$

$=$

$\ds \prod_{n \mathop = 0}^\infty \paren {1 + \frac {4 z^2} {\paren {2 n + 1}^2 \pi^2} }$

$\ds $

$=$

$\ds \paren {1 + \dfrac {4 z^2} {\pi^2} } \paren {1 + \dfrac {4 z^2} {9 \pi^2} } \paren {1 + \dfrac {4 z^2} {25 \pi^2} } \dotsm$

$\ds \cos z$

$=$

$\ds \prod_{n \mathop = 0}^\infty \paren {1 - \frac {4 z^2} {\paren {2 n + 1}^2 \pi^2} }$

$\ds \leadsto \ \ $

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

$=$

$\ds \prod_{n \mathop = 0}^\infty \paren {1 - \frac {4 \paren {i z}^2} {\paren {2 n + 1}^2 \pi^2} }$

$z \to i z$

$\ds \leadsto \ \ $

$\ds \map \cosh z$

$=$

$\ds \prod_{n \mathop = 0}^\infty \paren {1 + \frac {4 z^2} {\paren {2 n + 1}^2 \pi^2} }$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 38$: Infinite Products: $38.4$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 38$: Infinite Products: $38.4.$