A Complex Infinite Product

Using Euler's observation , we have that

$\displaystyle \sin(w) = w\prod_{k=1}^{\infty} \left(1-\dfrac{w^2}{\pi^2k^2}\right)$

Substituting $w = \dfrac{(1-i)\pi}{2}$ , and using the exponential formula for complex sine , the required product is

$z = \dfrac{\sin w}{w} = \left(\dfrac{e^\frac{\pi}{2}+e^{-\frac{\pi}{2}}}{2 \pi}\right)\cdot (1+i)$

Therefore $(1-i)z = \dfrac{2\cosh(\frac{\pi}2)}{\pi}$ and $\text{Arg}(z) = \frac{\pi}4$ giving the sum $2.382790$ .