Second reflection in function on curve: Ch 5, XI pp. 256-257

Gary

On page 1 of your document, the second equation from the bottom has the second term involving $|w|$, which you have left in place and solved
for $\bar{w}$ on the bottom line.

However, just as in the equations on page 256 for the Schwarz function and the Schwarzian reflection of $z$ across $K$, $z\bar{z}=|z|^2$ is only true for $z$ on $K$, so for the ellipse $E$ the Schwarz function is only equal to $\bar{z}$ if $z$ is on the ellipse.

So again $|w|=(w\bar{w})^{1/2}$ is only true on the ellipse. This means that generally (i.e. for points which could be on or off the ellipse), we should write $\tilde{w}=\mathfrak{R}_E(w)$ as on page 255, noting that if $w$ is on the curve then $\tilde{w}=\bar{w}$, and so we should also write your second equation as
$\tilde{w}+\alpha(w\tilde{w})^{1/2}+w+\beta=0$, to avoid confusion, where $\alpha$ and $\beta$ are constants depending only on $a$ and $b$, and you need to re-arrange this to solve for $(\tilde{w})^{1/2}$.

We can write $w\tilde{w}=w\bar{w}=|w|^2$ only if $w$ is on the ellipse.

I think the easiest way to do this is to substitute $q=(\tilde{w})^{1/2}$ and $\tilde{q}=(w)^{1/2}$ which gives a quadratic in $q$. Solving this quadratic for $q$ we can then calculate $\tilde{w}$ as $q^2$.

As before you will probably need to be careful choosing the sign of the square root!

Vasco

Gary

I agree with your comment about $|w|=(w\bar{w})^{1/2}$, and your other comments. I meant to write $|w|=(w\widetilde{w})^{1/2}$ ($w$ tilde not $w$ bar). However this correction is of no real consequence since as I understand from reading your post, my comments in general were not relevant to the problem as you see it.

I have looked again and the only thing I noticed is that at the bottom of page 1 of your document you seem to be wanting to map the original ellipse $E$ using the function $w=f(z)=z^2$. However in the subsequent maths you seem to have taken the equation of the ellipse $E$ and substituted for $z,\bar{z}$ to obtain an equation in $w$. This new equation in $w$ is not the equation of the ellipse under the mapping $f=z^2$ and so I think that as you had concluded yourself, your equation for $\mathcal{S}_f(w)$ is not correct.

Vasco

Gary

So the plot of the ellipse on the right should be correct, but if you are using the equation for any of the other calculations then it could obviously be a problem. I have derived the equation but I haven't checked it out yet.

Vasco

Gary

I have written a short document which is here showing how I derived the equation of the transformed ellipse. I'm not sure if it will be any use to you.
In the course of producing this document and plotting the transformed ellipse, I realised that the vertical semi-axis is not 3 but 2 which puts the ellipse in a stable position the way yours is, with the horizontal axis longer than the vertical axis. So my statements in red in the above quote are incorrect.

I certainly agree with your comment in a recent post that drawing a diagram is a good investment. This is in the spirit of Needham's book of course - Visual Complex Analysis!

Vasco

Vasco,

I followed up on your equation for the squared ellipse. It seems to work out well. Two new equations derived from it produced the same ellipse in the plot as the former, but there was a great difference in the Schwarzian function. The composite function $\mathfrak{R}_f \circ f \circ \mathfrak{R}_E(z)$ now seems to work as intended. See if you think it is what Needham had in mind. I know the plot is overly busy, but there were quite a few interdependencies in the problem.

nh.ch5.pp.256-257.pdf (241.9 KB)
(Revised May 29, 2016)

By the way, there is a typo in your document, paragraph 3: Should be  $1 - 2 sin^2 t = cos 2t$. It wasn't carried through.

Gary

Gary

The way I see the ideas on page 257 is that they are the full generalisation of Schwarz's Symmetry Principle, which subsection 5 of section XI slowly introduces, building slowly from 'simple'(?) ideas to the general Schwarz Principle.

The way I understand your document is as follows:

The curves $L$ and $\widehat{L}$ in the particular case you have investigated are the segments of the two ellipses. The function $f$ is $z^2$ and $\mathfrak{R}_L$ and $\mathfrak{R}_{\widehat{L}}$ are the Schwarzian reflections in the two ellipses.

The function $f^{\ddagger}$ is the analytic continuation of $f=z^2$ from one side of $E$ to the other, and so should turn out to be $z^2$ on this extended region. The example you have chosen is an example like the one described on page 254 in the second paragraph which begins:
"Of course if $f$ is already defined...". I would be interested to know if $f^{\ddagger}$ is indeed behaving like $z^2$.

In your document, at the end, where you list problems you have identified, it seems to me that problem 2 is not really a problem. Generally speaking the Schwarzian Reflection function $\mathfrak{R}$ only approximates to normal reflection. The closer the point is to the curve (in your case the ellipse), the more accurately the Schwarzian reflection coincides with normal reflection. So I don't think it is surprising that $f(a)$ and $f\circ\mathfrak{R}_E(a)$ do not quite lie on the circle.

Vasco

Gary

I have looked again at your document and apart from the typo of "a=1 and b=1" on page 3 just before your formula for $\mathcal{S}_f(w)$, it all seems to tie together very well. I have used a spreadsheet to check the formula for $\mathcal{S}_f(w)$ and it seems spot on. Do you still have any concerns about it?

Vasco