I am confused by the following two approaches
\begin{align} \frac{i}{2}\left(\log(ia+w)-\log(ia-w)\right)&=\frac{i}{2}\log\frac{ia+w}{ia-w}\\ &=\frac{i}{2}\log\frac{(ia+w)^2}{\color{red}{i^2}(a^2+w^2)}\\ &=i\log\frac{w+ia}{\color{red}{i}\sqrt{a^2+w^2}}\\ &=i\log\frac{a-iw}{\sqrt{a^2+w^2}}\\ &=i\log e^{-i\tan^{-1}\frac{w}{a}}=\tan^{-1}\frac{w}{a} \end{align} Mistake in red fixed!
\begin{align} \frac{i}{2}\left(\log(ia+w)-\log(ia-w)\right)&=\frac{i}{2}\left[\log\left(i(a-iw)\right)-\log\left(i(a+iw)\right)\right]\\ &=\frac{i}{2}\left[\frac{i\pi}{2}+\log(a-iw)-\frac{i\pi}{2}-\log(a+iw)\right]\\ &=i\log\left[\frac{a-iw}{\sqrt{a^2+w^2}}\right]\\ &=i\log e^{i\tan^{-1}\frac{-w}{a}}=\tan^{-1}\frac{w}{a} \end{align}