z^{1/2}

Here is a picture of the function z^{1/2} on the domain C\(-infty,0]. It is a "branch" of the inverse to the function z^2 with value 1 at z = 1. Notice that only half of the spectrum of colors occurs as you circle around the origin. Can you see how the square root opens the slit plane to a half plane? Also the the shading is lighter near 0 and darker for large z than the standard coloring of the plane, though it may be hard to see. If you look carefully, you can see that the color sectors are twice as wide and the shading annuli have been changed in size, when compared with the standard coloring.

Here is a picture of the function z^{1/2} on the domain C\[0,+infty). It is the other "branch" of the inverse to the function z^2 with value -1 at z = 1. Notice how the two pictures "fit" together along the negative real axis. It is possible to make a surface in R^4 by cutting each "sheet" along the negative reals then attaching the sheets together along the edges of the cuts, so that the coloring changes continuously along the surface. (Top edge of cut in top picture to bottom edge of cut in bottom picture, bottom edge of cut in top picture to top edge of cut in bottom picture)