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I have another problem that has eluded me for days and I'm sure I'm close. If anyone can help, please nudge me in the right direction.

Consider the mapping w = u + iv = 1/z, where z = x + iy. Show that the region between the curves v = -1 and v = 0 maps into the region outside the circle x^2 + (y - 1/2)^2 = 1/4 and above the line y = 0.

I know that w = x/(x^2 + y^2) + i[-y/(x^2 + y^2)].

I also figured since at v = -1, in the x-y plane, y/(x^2 + y^2) = 1.

Alas I know not where to go from here. I have attatched a sketch of the region in the v-u plane.

Thanks in advance.

PS: the question then goes on to ask :- What is the image in the x-y plane of the line -1/2? This is obviously a circle since a straight line in one plane is a curve in the other, but how do I prove this?

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# More Complex Complex Analysis

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