Find the image of the of the given set under omega z2 a x y

Find the image of the of the given set under omega = z^2. (a) {(x, y):y = x) (b) {(x, y):1

Solution

we know z = x + i y

z²  = ( x + i y )² = x² - y² + i 2xy

a) when x = y then the real part becomes and imaginary parts become 2x² or 2y²

the images of w has only imaginary which is twice the square of given input values

b) as X lies between 1 and 2 the value of y which is not given then the image has both real and imaginary parts which includes the value of y as mentioned above.