Question

Find the image of the half plane y>1 under the transformation w=(1-i)z

Answer 1

Answer on Question #67556 – Math – Complex Analysis

Question

Find the image of the half plane $y > 1$ under the transformation $w = (1 - i)z$.

Solution

$$z = x + iy,$$

where

$$y > 1$$ $$w = (1 - i)z = (1 - i)(x + iy) = x - ix + iy + y = x + y + i(y - x)$$ $$w = u + iv,$$

where

$$\left\{ \begin{array}{l} u = x + y \\ v = y - x \end{array} \right.$$

Add the first and the second equations of the system

$$u + v = x + y + y - x = 2y > 2,$$

because

$$y > 1.$$

The image will be the half plane $u + v > 2$.

If we assume that $w = x + iy$, the image will be the half plane $x + y > 2$.

**Answer**: If we assume that $w = x + iy$, the image will be the half plane $x + y > 2$.

Answer provided by https://www.AssignmentExpert.com