COMPLEX ANALYSIS:
Can somebody please thoroughly explain to me why the solution to
sin(z) = w
is
z = -i*log (i*w + (1-w^2)^(1/2))
and not
z = -i*log (i*w ± sqrt(1-w^2))
If $\displaystyle sin(z) = w$ then $\displaystyle cos(z) = \sqrt{1-w^2}$ not $\displaystyle \pm \sqrt{1-w^2}$ by the Pythagorean theorem.
So, you have:
$\displaystyle e^{iz} = cos(z) + i w$
$\displaystyle e^{iz} = iw + \sqrt{1-w^2}$
$\displaystyle iz = \log{(iw + \sqrt{1-w^2})}$
$\displaystyle z = -i \log{(iw + \sqrt{1-w^2})}$