Complex number transformations involving arguments

@Zacken @TeeEm

Can you be more specific? I don't fully understand your question. Where does 0 < x < 1 and -1 < y < 0 come from?

So instead of treating arg(z) = pi/4 as x + xi the MS suggested, I did it the argument way and got arg (1 - wi) - arg (w -1) = pi/4, which is a circle that satisfies mod w = 1, however only within the ranges I posted above. Surely the MS method is wrong as it treats arg (z) = pi/4 as x + xi, which is only true for values in the positive quadrant.. hope that makes more sense..?

But arg(z) = pi/4 is only values in the positive x and y quadrant. That's why it's called a half-line.

Anyways, with your method, why does arg(1-wi) - arg(w-1) bring up the ranges that you specified?

Does that help?

Question doesn't say the image of the given half line is the circle, only that the image of the half line lies on the circle - this doesn't imply the entire circle.

I realise that, but am I right in saying that the complex numbers on arg(z) = pi/4 can only be mapped onto those in the shaded region I previously specified?

Yes, it's just the lower right quadrant, but it's the part of the circle only, not any area.