Problem 1521 (difficulty: 3/10)

Let \(\displaystyle w(z)=\frac 12\left(z+\frac 1z\right)\) be the so called Zhukowksy map. What is the image of

(a) the unit circle? (b) the interior of the unit circle? (c) the exterior of the unit circle?

(d) the circles with center \(\displaystyle 0\)? (e) the lines passing through \(\displaystyle 0\)?

Answer:

(a): The line segment \(\displaystyle [-1,1]\).

(b) and (c): The complement of \(\displaystyle [-1,1]\).

(d): Ellipses with foci \(\displaystyle -1,1\). (The unit circle is mapped to the line segment \(\displaystyle [-1,1]\).)

(e): Hypebolas with foci \(\displaystyle -1,1\). (The image of the real axis is the union of the rays \(\displaystyle (-\infty,-1]\) and \(\displaystyle [1,\infty)\); the imaginary axis is mapped onto itself.)

Give me another random problem!