Images of complex function

[Steven G]

I just paid my admission fee for help, by helping with posted problems.
Suppose w=f(z)= (i-1)/(z+i).
How can I find the image of
a) |z|=1 (the unit circle)
b) |z|<1 (inside the unit circle)
c) the imaginary axis.

Of course I can plot points that that is not the way to graph
For c) I got the the image was the line that crosses 1+i and -1-i but does not include the point 0

 

[mario99]

What have you done so far for (a)?

Hint: Parametrize the curve and let Desmos draw it.

 

[mario99]

For (a), I got this line. I am wondering if professors Dave and blamocur will get something else!

 

[mario99]

View attachment 38653

For (a), I got this line. I am wondering if professors Dave and blamocur will get something else!

This is for (b) if my calculations were correct ([imath]r = 0.7[/imath]). Both results, (a) and (b) are expected because the image of circles under [imath]\displaystyle\frac{i - 1}{z + i}[/imath] are either lines or circles.

The result of (c) is more interesting than (a) and (b). I will leave it for you as an exercise.