for {${z\in \mathbb C : Im(z)>0}$}, we simply sketch the upper half of the Real axis, right?
Then, if we have $z=a+ib$, and we sketch that, and we have $w=iz=-b+ai$ which means $w+1=(1-b) + ai$
How does the sketch of $w$ tranform to $w+1$
Simple question it seems, but does it move to the right by one, or to the left? I though it was to the left since $f(x) \to f(x+1)$ shifts by one in the $x$ direction to the left...
Any help? Thanks
I find it easiest when starting off to understand the argand diagram in terms of coordinates.
Given a point $z = a+ib \in \mathbb{C}$ it lies in the plane at coordinates $(a,b)$.
For example $1$ is the point $(1,0)$, $i$ is the point $(0,1)$, and $1+4i$ is just the point $(1,4)$.
We also have $a = \text{Re}(z), \ b = \text{Im}(z)$.
Now $$\{ z \in \mathbb{C} : \text{Im}(z) > 0 \} = \{ a+bi \in \mathbb{C} : b > 0 \} = \{ (a,b) : b > 0 \}$$
This looks like the "top" half of the plane, above the real axis:
(Note also in this picture where $1+i$ is in relation to $i$)
Now if $z$ is the $(a,b)$, can you see where $iz = (-b,a)$ ends up in the plane?
Maybe try it with a few examples, and take a guess at the pattern. See what happens when you times by $i$ a few more times, does it get back to where it started?
Hopefully with coordinates it's easier to see where $w+1$ is in relation to $w$.