Q: Find the image of the infinite strip under the transformation .
(This question is taken from Complex Variables and Applications (Brown and Churchill))
Answer: , .
We are using the standard notation , . From the equation , we can conclude that .
With some algebra and completing the square, one can obtain , which is the equation of a circle centered at with radius . When , the equation of the circle is . As gets smaller (closer to zero), the radius of the circle becomes larger, while still remaining tangent to the horizontal axis.
Thus, the image of the strip is , .