Image of a circle under w = 1/z

The circle |z − i| = 1 passes through the origin, so under w = 1/z it maps to a straight line (a circle through the origin maps to a line).

Let z = x + iy. The circle is x² + (y − 1)² = 1, that is

x² + y² = 2y        ... (1)

Now w = 1/z = z̄/|z|² = (x − iy)/(x² + y²). So

u = x/(x² + y²),   v = −y/(x² + y²)

Using (1), x² + y² = 2y, so

v = −y/(2y) = −1/2

Answer: the image is the straight line v = −1/2 (the line Im(w) = −1/2, parallel to the real axis). The point z = 0 on the circle goes to w = ∞, so the whole line is covered. Check: z = 2i gives w = −i/2, and z = 1 + i gives w = (1 − i)/2; both have v = −1/2.