# Hello

## 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) |

**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.