a) Let C be the positively oriented simple closed contour consisting of the four sides of the rectangle with vertices at $2, 3, 3 + 4i $ and $2 + 4i$. Is it true that $$|\begin{equation} \int_{C} \frac{ (\bar z^3 -10) e^{iz}}{Log z} dz \end{equation}| < 2000 ?$$ b) Let $γ$ be the positively oriented rectangle with vertices at $−2 + i, 2 + i, 2 + 4i, −2 + 4i$. Is it true that \begin{equation} \int_{γ} \frac{ z- Log z}{z+Log z} dz =0 \end{equation}
This is what i have done so far:$$\\$$ a) Using the ML inequality $$|\begin{equation} \ \frac{ (\bar z^3 -10) e^{iz}}{Log z} \end{equation}| \le \frac{(|z^3| -|10|) |e^{iz}|}{ln |z|} \le \frac{(5^3 -10)(e^{-4})}{ln 5} = M $$ $$\\$$ With $ \\ ln|z| \le ln |3+4i| = ln \sqrt 25 = ln 5 \,\ , |e^{iz}|= |e^{i(3+4i)}| =e^{x} = e^{-4} ,\ L = 10$ $$|\begin{equation} \ \frac{ (\bar z^3 -10) e^{iz}}{Log z} \end{equation}| \le M L = 13.071.. < 2000 $$ $$\\$$
b)$Re (Log z) = ln|z| \le |2+4i| = ln \sqrt20 > -2 \ge Re (z)$ $ Since \ z+Log Z \ne 0 \ \text{and Log is not analytic only on } (-∞,0]$\ by Cauchy Goursat Thm \begin{equation} \int_{γ} \frac{ z- Log z}{z+Log z} dz \ne 0 \end{equation}