Mathematical Tools for Physics

14—Complex Variables 439

0 1 2

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LogarithmHow about a logarithm? lnz= ln

(

reiθ

)

= lnr+iθ. There’s a branch point at the origin, but this
time, as the angle keeps increasing you never come back to a previous value. This requires an infinite number of
sheets. That number isn’t any more difficult to handle — it’s just like two, only bigger. In this case the whole
winding number around the origin comes into play because every loop around the origin, taking you to the next
sheet of the surface, adds another 2 πiw, andwis any integer from−∞to+∞. The picture of the surface is
like that for the cube root, but with infinitely many sheets instead of three. The complications start to come
when you have several branch points.

Two Square RootsTake

√

z^2 − 1 for an example. Many other functions will do just as well. Pick a base point
z 0 ; I’ll take 2. (Not two base points, the number 2.)f(z 0 ,0) =

√

3. Now follow the function around some loops.
   This repeats the development as for the single branch, but the number of possible paths will be larger.

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z 0

Despite the two square roots, you still need only two sheets to map out this surface. I drew theaband
cd cuts below to keep them out of the way, but they’re very flexible. Start the base point and follow the path
around the point+1; that takes you to the second sheet. You already know that if you go around+1again it
takes you back to where you started, so explore a different path: go around− 1. Now observe that this function
is theproductof two square roots. Going around the first one introduced a factor of− 1 into the function and