Problem 6 Part5

Hi Alan

Yes there is. I first did this about 15 years ago and used something called PSTricks for the graphics. Nowadays there are quite a few new and easier ways.
In my answer at the bottom of page 5 there is an equation in $x$ and $y$ for a given value of $m$ and this when plotted with different values for $m$ will give you the blue curves.
To obtain the red curves you need to use a similar argument to the one on page 5, used to find the result at the bottom of the page:
The equation of a circle centred at $w=1$ with radius $R$ is $(u-1)^2+v^2=R^2$.
Since $u=x^2-y^2$ and $v=2xy$ we can substitute for $u$ and $v$ and obtain the equation
$[(x^2-y^2)-1]^2+4x^2y^2=R^2$
If we plot this for different values of $R$ we obtain the red curves.
The easiest way to do the plots is to use Desmos.
Here is a link to my plot:
www.desmos.com/calculator/fwe2vbw0kl
This shows you also how it is done, and as you can see it is very straightforward. The important thing is to do the mathematics on page 5. You can explore Desmos in your own time. It is a very useful graphics application and completely free. It can do very simple graphics but also more complex graphics as you can see from this example.
If you have any more questions about the maths or about Desmos please ask me.

Vasco

Hi Alan

When I received your post I concentrated on reading part (v) of my answer. After reading your latest post and the original exercise I decided to read my answer to the whole exercise and I think that you are right that part (v) should be done in another way.
I will take another look and publish my amended answer. Thanks

Vasco

Hi Alan

I have just updated my document to change the layout slightly.

Vasco