**1) f(z)=z^{2}:** Try some circles centered at the origin. Can you
explain the spacing of the image circles? Try some
rays from the origin. Then trace a semicircle like curve from a point on
the positive real axis to one on the negative real axis. Do you
understand what you see?

**2) f(z)=exp(z):** Uisng the tool at the lower left
of the Complex Paint palette, look at the image of a collection of
small rectangles contianing the origin.
Do you see the relation to polar coordinates?

**3) f(z)=log(z):** Try a fairly big circle centered at the origin.
Then one with a small radius. Perhaps you need to select the menu entry

**4) f(z)=sqrt(z):** Try some rays from the origin. Then a box crossing the
negative real axis. Then a box crossing the positive reals. Where is the
branch cut for this function?

**5)
f(z)=exp(log(z)/2):** Note this is a branch of the square root
function. Is the branch the same as sqrt(z) above?

**6)
:** Try the image of a
circle of radius 1 about the origin. Where does 0 go? What does this suggest
about the image of the unit disk? Now try a diameter of the unit circle
.
Where does it go? Is the image perpendicular to the unit
circle? Straight lines and circles orthogonal to the unit circle are
sometimes used as models for lines in *hyperbolic geometry*.

**7)
:** Inversion.

**8)
f(z)=sqrt(1-z):** Can you explain the behavior of this branch?