Image of Infinite Strip under Transformation w=1/z

Q: Find the image of the infinite strip $0<y<1/(2c)$ under the transformation $w=1/z$ .

(This question is taken from Complex Variables and Applications (Brown and Churchill))

Answer: $u^2+(v+c)^2>c^2$ , $v<0$ .

Solution:

We are using the standard notation $w=u+iv$ , $z=x+iy$ . From the equation $\displaystyle z=\frac{1}{w}=\frac{1}{u+iv}=\frac{u-iv}{u^2+v^2}$ , we can conclude that $\displaystyle y=\frac{-v}{u^2+v^2}$ .

With some algebra and completing the square, one can obtain $u^2+(v+\frac{1}{2y})^2=(\frac{1}{2y})^2$ , which is the equation of a circle centered at $(0,-\frac{1}{2y})$ with radius $1/2y$ . When $y=1/(2c)$ , the equation of the circle is $u^2+(v+c)^2=c^2$ . As $y$ gets smaller (closer to zero), the radius of the circle becomes larger, while still remaining tangent to the horizontal axis.