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I'm solving the following exercise:

Use the estimate lemma to prove that $$\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 4\sqrt{10},$$where $\gamma$ is the square with vertices $\pm 1 \pm i$.

Clearly the lenght of the square is $8$. Starting from $-1-i$, counterclockwise, call the sides of the square $\gamma_1,\ldots,\gamma_4$.

  • For $\gamma_1$ and $\gamma_3$, we have $|z-2| \leq \sqrt{10}$ and $|z-3| \geq \sqrt{5}$, whence: $$ \left|\frac{z-2}{z-3}\right|\leq \frac{\sqrt{10}}{\sqrt{5}} = \sqrt{2}. $$

  • For $\gamma_2$, we have $|z-2| \leq \sqrt{2}$ and $|z-3| \geq 2$, so: $$\left|\frac{z-2}{z-3}\right|\leq \frac{\sqrt{2}}{2}. $$

  • For $\gamma_4$, we have $|z-2| \leq \sqrt{10}$ e $|z-3| \geq 4$, so: $$\left|\frac{z-2}{z-3}\right| \leq \frac{\sqrt{10}}{4} $$ The greatest of these upper bounds is $\sqrt{2}$. So that inequality is good on all of $\gamma$. We get:$$ \left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 8\sqrt{2}. $$

I got these inequalities geometrically, looking at maximum and minimum distances from $2$ and $3$ to said curves $\gamma_j$. I don't see how he got $4\sqrt{10}$. Can someone help me please?

Here's a figure to make your life easier:

enter image description here

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    $\begingroup$ Since $8\sqrt{2} < 4\sqrt{10}$, you have a stronger bound. $\endgroup$ – JimmyK4542 Jun 20 '15 at 3:35
  • $\begingroup$ I thought that for a second there, but I got insecure. So far the book had given the best bounds. But I guess you're right: $$8\sqrt{2} < 4\sqrt{10} \iff 64 \cdot 2 < 16 \cdot 10 \iff 128 < 160 \quad \checkmark$$ $\endgroup$ – Ivo Terek Jun 20 '15 at 3:38
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    $\begingroup$ Yes, @JimmyK4542 beat me to it (+1). You have shown the desired result. $\endgroup$ – MPW Jun 20 '15 at 3:38
  • $\begingroup$ Thanks guys. I thought I was crazy. I'll delete the question then. :) $\endgroup$ – Ivo Terek Jun 20 '15 at 3:38
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    $\begingroup$ I'll make a CW answer to wrap up this. $\endgroup$ – Ivo Terek Jun 20 '15 at 3:40
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I'll convert my comment into an answer:

Since $8\sqrt{2} < 4\sqrt{10}$, you have found a stronger upper bound for $\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right|$ than what was required by the problem.

We can get the weaker upper bound that the problem is asking for as follows:

For all $z$ on the curve, we have $|z-2| \le \sqrt{10}$ and $|z-3| \ge 2$.

Hence, $\left|\dfrac{z-2}{z-3}\right| \le \dfrac{\sqrt{10}}{2}$, and thus, $\displaystyle\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \le 8 \cdot \dfrac{\sqrt{10}}{2} = 4\sqrt{10}$.

This gives us a weaker upper bound since we didn't have to consider 3 seperate cases.

We can get an even stronger upper bound by using the Residue Theorem to get $\displaystyle\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| = 0$, (because the only pole $z = 3$ is outside $\gamma$), but that's overkill.

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  • $\begingroup$ I knew that it was supposed to be simple! Thanks again - not only my answer was also correct, but you even indicated how to get the proposed bound. Very helpful, thanks again. $\endgroup$ – Ivo Terek Jun 20 '15 at 4:04
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In fact, I got a better bound than the one proposed, so there's no problem at all: $$8\sqrt{2} < 4\sqrt{10} \iff 64 \cdot 2 < 16 \cdot 10 \iff 128 < 160 \quad \checkmark$$

Thanks Jimmy and MPW for the useful comments.

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