Exercise 14, Chapter 7

Vasco,

I think it bakes up well. In mine, I have "On the first half of the domain, h(z) is zero, so h(z) is a line of length r on the real axis between 0 and (1/2)," which is a non-sequitor. It should read "On the first half of the domain, h(z) is zero, so h(z) is a point or the transposition of many points as r varies; r is plotted as a thick black line of length r on the real axis between 0 and (1/2)". As for part (ii), I missed the point that there are an infinite number of p-points associated with the shrinking of r. The pizza metaphor really does help to see this. I will fix my answer later today.

Gary

Gary

It seems clear to me from the meaning of $p$-points, as explained on page 345, and the definition of $h$ given in the exercise, that when $h=0$ its preimages are all the points $0\leq z<(1/2)$ lying inside the circle $|z|<(1/2)$, since all these points map to $h=0$. This could be rephrased as being true by definition. Also, since $\Gamma$ is the circle $|z|\leq (3/4)$, all these $p$-points are located inside $\Gamma$.

The comma is in the place I intended, but I can see that it might confuse and so I would suggest three ways to re-write it:

1) If we label as $\Gamma$ the dough lying on the circle $|z|=(3/4)$, on the original pizza, then we can see that its new position, (after we stretch the pizza according to the mapping $h$), is on the circle $|z|=(3/4)[(2(3/4)-1]=(3/8)$.

2) If we label as $\Gamma$ the dough lying on the circle $|z|=(3/4)$, on the original pizza, then we can then see that, after we stretch the pizza according to the mapping $h$, its new position is on the circle $|z|=(3/4)[(2(3/4)-1]=(3/8)$.

3) Let's label as $\Gamma$ the dough lying on the circle $|z|=(3/4)$, on the original pizza. Now stretch the pizza according to the mapping $h$.   We can now see that its new position, is on the circle $|z|=(3/4)[(2(3/4)-1]=(3/8)$.

Vasco

Vasco,

I like 3), but remove comma after "its new position".

Gary