On page 135 of Joerg Bruedern's "Einfuehrung in die analytische Zahlentheorie" he claims that Stirling's formula implies for fixed $\sigma <0$, any $t\geq 1$, and some constant $C$ (I assume depending on $\sigma $ but not on $t$)
$$\Delta (\sigma +it)=Ce^{-it\log t+it(1+\log 2\pi )}t^{1/2-\sigma }+\mathcal O(t^{-1/2-\sigma }),$$
where $\Delta (s)$ is "the factor in the functional equation for the Riemann Zeta Function", which I assume means
$$\Delta (s)=\frac {\pi ^{s-1/2}\Gamma ((1-s)/2)}{\Gamma (s/2)}.$$
Stirling's formula says that for fixed $\delta >0$ and $\text {arg}(z)\leq \pi -\delta $
$$\log \Gamma (z)=(z-1/2)\log z-z+\frac {\log 2\pi }{2}+\mathcal O(1/|z|).$$
How does the claim follow from this? I've tried to expand $z^{z-1/2}$ by writing it in terms of the real and imaginary parts of $z$ and using the definition of complex powers, but my answer always seems far from a simple formula like above. In particular I always have terms of the form
$$e^{i\text { arg}(z)}$$
which I don't know what I should do with.