Answer to Question #91404 in Complex Analysis for Sajid

Question

Choose the correct answer.
Q. Which of the following is image of straight line x=1 under the transformation f(z)= z^2, where z=x+iy?
a. A circe
b. An ellipse
c. A parabola
d. A hyperbola

Answer 1

ANSWER: (c).

The line x=1 is transformed into a parabola with the mapping f(z)=z² . (z=x+iy)

Explanation:

Denote w=f(z)=z² =u+iv. f(1+iy)= (1+iy)² =1-y²+2iy. Determining the real and imaginery

part of w, we obtain the parametric equation of the image of a line x=1: u=1-y² , v=2y.

Excluding the parameter «y» , we get the u=1-v²/4 (a parabola).