Comoplex Analysis?

Question

I need help drawing the contours of f(z)=1/z. I've found Re f =u(x,y) = x/(x^2+y^2) and Im f = v(x,y) = -y/(x^2+y^2). Now what do I do?
Please include an explanation. Thanks...

Answer 1

To find the contours put
x/(x^2+y^2)=k .For each k you get
x^2+y^2-1/k*x=0 circunferences center(1/2k,0)
you should exclude (0,0)
The same for
x^2+y^2+1/k*y=0 Center (0,-1/2k)