Mathematical Tools for Physics

14—Complex Variables 438

are. Sideais the same as sidea. The same forb. 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); a
secondis 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 I defined 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 Riemann surface. Again, I’ll draw a closed loop. As long as you
have only a single branch point it’s no more complicated than this.