ML estimate problem

Question

I get: $$R\frac{\pi}{e^{-y}}|(z^2+1)|$$

What am I doing wrong?

Answer 1

Note the following inequality:

$$|z^2| = |(z^2 + 1) - 1| \leq |z^2 + 1| + 1 \implies |z^2 + 1| \geq |z^2| - 1$$

So we have:

$$ I =\left | \int_C \frac{e^{iz}}{z^2 + 1}dz \right| \leq \int_C \frac{\left | e^{iz} \right |}{\left| z^2 + 1 \right|}|dz| = \int_C \frac{e^{-y}}{\left| z^2 + 1 \right|}|dz| $$

We are given that $\text{Im}(z) \geq 0$ so $e^{-y} \leq 1$ Using this and the first inequality I derived we get:

$$ I \leq \frac{1}{R^2 - 1} \int_C 1\cdot|dz| = \frac{\pi R}{R^2 - 1} $$