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Use the ML estimate to check that $|\int_\gamma e^z -\bar{z}| \leq 57$ where $\gamma$ is the boundary of the triangle with vertices at $0, 3i, -4$.

I have that $L = 12$, the length of the triangle. I am having trouble figuring out M so that ML = 57. I was trying to follow an example that my professor gave us.

By the triangle inequality $|e^z -\bar{z}| \leq |e^z| + |\bar{z}|$. Then $|e^z|= e^x \leq 1 $ since $-4 \leq x \leq 0$. Also $|\bar{z}| = \sqrt{x^2 + y^2} \leq 5$. So by my estimate $ |\int_\gamma e^z -\bar{z}| \leq 12(1+5) = 72$. Where is my thinking going wrong? I'm not sure how we are suppose to come up with 57?

frostyfeet
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1 Answers1

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You don't have to sharpen the estimate too much - just be more careful than the trivial estimate. One first, easy thing to note is that $|\overline{z}| \le 4$ always - look at the triangle. This already improves the estimate from $72$ to $60$.

You can improve further by considering segments separately. On the vertical part from $0$ to $3i$, you have $|\overline{z}| \le 3$ instead of $4$. This already saves you the $3$ necessary to go from $60$ to $57$ and is good enough.

Of course, $57$ is far from the optimal estimate. You could play this game on smaller intervals too. On the segment $[0, i]$, you have the estimate $|e^z + \overline{z}| \le 2$, which gets us to $55$, and on $[i, 2i]$ the estimate $|e^z + \overline{z}| \le 3$ gets us to $54$. You could keep going if you'd like.