Infinite Product of One Plus Reciprocals of Squares
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Theorem
| \(\ds \prod_{n \mathop = 1}^\infty \paren {1 + \frac 1 {n^2} }\) | \(=\) | \(\ds \paren {1 + \frac 1 1} \times \paren {1 + \frac 1 4} \times \paren {1 + \frac 1 9} \times \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh \pi} \pi\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^\pi - e^{-\pi} } {2 \pi}\) |
Proof
| \(\ds \prod_{n \mathop = 1}^\infty \paren {1 + \frac 1 {n^2 } }\) | \(=\) | \(\ds \prod_{n \mathop = 1}^\infty \frac {n^2 + 1} {n^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \prod_{n \mathop = 1}^\infty \frac {\paren {n - i} \paren {n + i} } {\paren {n - 0} \paren {n - 0} }\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \Gamma 1 \map \Gamma 1} {\map \Gamma {1 + i} \map \Gamma {1 - i} }\) | Infinite Product of Product of Sequence of $n + \alpha$ over Sequence of $n + \beta$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \Gamma 1 \map \Gamma 1} {i \map \Gamma i \map \Gamma {1 - i} }\) | Gamma Difference Equation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {0! \times 0! \map \sin {i \pi} } {i \pi}\) | Gamma Function Extends Factorial, Euler's Reflection Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {i \sinh {\pi} } {i \pi}\) | Hyperbolic Sine in terms of Sine, Factorial of Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh {\pi} } \pi\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^\pi - e^{-\pi} } {2 \pi}\) | Definition of Hyperbolic Sine |
$\blacksquare$