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Question 6

Find the image of the unit square $0\leq x\leq 1, 0\leq y\leq 1,$ under the mapping $w=(1+i)z$.

A) Use the (x,y) coordinates of points A,B,C, & D on the z-plane, use their coordinates (x,y) to find the coordinates of the mapping of the w-plane:

$z=x+iy \rightarrow w=(1+i)z = (1+i)(x+iy)=x+iy+ix=i^2y=x+iy+ix-y$
$w=x+iy+ix-y$
{u=x-y}
{v=y+x}

Now to find the w-plane coordinates, use the u and v equations found above.

Coordinate Name Z-Plane Algebra W-Plane
A (0,0) (0-0,0+0) (0,0)
B (0,1) (0-1,0+1) (-1,1)
C (1,0) (1-0,1+0) (1,1)
D (1,1) (1-1,1+1) (0,2)
HW%20Geo.png

B) Describe the overall effect of the mapping:
The mapping resulted in the figure from the z-plane to be rotated 45 degrees counter-clockwise and the range of the vertical axis changes from {0,1} to {0,2} and the horizontal axis from {0,1} to {-1,-1}.
$\therefore$ The z-plane figure's domain and range were both doubled and the area was doubled and rotated 45 degrees counterclockwise.

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