Q: Find the image of the infinite strip under the transformation .

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

Answer: , .

Solution:

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 , .

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