1) f(z)=z2: 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
Type Domain and Range on the Equation menu to adjust the range
so you can actually see the results. Can you explain the shape and location
of the images? What branch cut does log(z) use in Complex Paint?
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?
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?
: 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.
f(z)=sqrt(1-z): Can you explain the behavior of this branch?