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complex function. (rotation)

Hi,

A function H(z) =e^(iθ)z generate a rotation mapping. Sketch the image of the semidisk |Z|<=2, Im Z <= 0 under H when

a.θ= pi/4

b.θ= -pi/4

c.θ= 3pi/4

ok, so basically my question for a. is, do i apply the constraints and draw the disk, then do the rotation, or do the constraints still apply after the rotation. If you look at what i have done in the attachment you will probably know what i'm asking, should the part shaded black with the '?' be included or not? Thanks.

Re: complex function. (rotation)

You substitute in the $\displaystyle \theta $ first and then find the image of the region under $\displaystyle H$. There's no other way. Yes, the region with question mark is included. You're just rotating the region counterclockwise by $\displaystyle \pi/4$. Part (b) will be a clockwise rotation by $\displaystyle \pi/4$, etc.