# Our second field expedition

Our first field expedition was not without adventures. What kind of adventures they were – this can be now only guessed looking at some cryptic comments. They refer to the text that does not exist any more. I have changed the example. Last night Goddess talked to me in my dream, and She suggested a different avenue from that that I have had in my mind before. I obey. And so the second example follows. Very similar to the previous one, just a little been different.
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Last time I considered the one parameter group of transformations defined, on the complex plane, by
$f_t:z\mapsto \frac{\cosh(t)z-i\sin(t)}{i\sinh(t) z+\cosh(t)},$

where $t\in\mathbf{R}.$

Today I propose a variation:

$g_t:z\mapsto \frac{\cosh(t)z+\sinh(t)}{\sinh(t) z+\cosh(t)}.$

Again, with some little effort (perhaps easier than it was before, because now there is no more imaginary $i$ explicitly in the formula) we can verify that

$g_{t+s}(z)=g_t(g_s(z))$ and $g_0(z)=z.$

In order to obtain the vector field from this “flow of complex numbers” I am calculating the derivative:

$Y(z)=\frac{d}{dt} g_t(z)|_{t=0}.$

I used Mathematica to get

$Y[z]=1-z^2.$

To draw this vector field on the $(x,y)$ plane I write $z=x+iy,$ and calculate
$1-(x+iy)^2=1+y^2-x^2-2ixy.$
So

$Y(x,y)= (\mathrm{Re} (Y(z)),\mathrm{Im}(Y(z))=(1+y^2-x^2,-2xy).$

Here is the vector field, and stream lines from this second field expedition

Of course there will also the third expedition. After that the mystery will slowly be be revealed.

## 4 thoughts on “Our second field expedition”

1. Bjab says:

Arku,
na Boga, z jakich annałów wyciągnąłeś ten pierwszy wzór z plusem w liczniku?

2. Bjab says:

Of course there will also the third expedition. After that the mystery will slowly revealed. ->
be be