14—Complex Variables 361b. When you have more complicated surfaces, arising from more complicated functions of the complex
variable with many branch points, you will have a fine time sorting out the shape of the surface.
0
1
(z 0 ,1)
(z 0 ,0)
(z 0 ,0)
(z 0 ,1)
b
a
a
b
I drew three large disks on this Riemann surface. Oneis entirely within the first sheet (the first
map); asecondis entirely within the second sheet. Thethirddisk straddles the two, but is is nonetheless
a disk. On a political map this might be disputed territory. Going back to the original square root
example, I also indicated the initial point at which to define the value of the square root,(z 0 ,0), and
because a single dot would really be invisible I made it a little disk, which necessarily extends across
both sheets.
Here is a picture of a closed loop on this surface. I’ll probably not ask you to do contour integrals
along such curves though.
0 1
a
b
b
a
Other Functions
Cube RootTake the next simple step. What about the cube root? Answer: Do exactly the same
thing, except that you need three sheets to describe the whole. Again, I’ll draw a closed loop. As long
as you have a single branch point it’s no more complicated than this.