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Worked Example
Branch Cuts for Multiple Branch Points
What branch cuts would we require for the function
f (z) = log
z − 1 z + 1
It is clear that there are branch points at ±1, but we have a non-trivial choice of branch cuts. Define z − 1 = r 1 eiθ^1 and z + 1 = r 2 eiθ^2 , as shown in the following diagram.
The most straightforward choice is to take two branch cuts, one emanating from each branch point to infinity. In the case shown, we choose 0 6 θ 1 < 2 π and −π < θ 2 6 π, and the consequent single-valued definition of f (z) is
f (z) = log(z − 1) − log(z + 1) = (log r 1 + iθ 1 ) − (log r 2 + iθ 2 ) = log(r 1 /r 2 ) + i(θ 1 − θ 2 ).
The two cuts make it impossible for z to “wind around” either of the two branch points, so we have obtained a single-valued function which is analytic except along the branch cuts.
The second possible choice is to take only one branch cut, between −1 and 1, as shown. This time, we choose both 0 6 θ 1 < 2 π and 0 6 θ 2 < 2 π (note that this seems at odds with the location of the branch cut, but this is not a problem as we will explain). The definition of f (z) is as before, but with these different ranges for θ 1 and θ 2.
Mathematical Methods II Natural Sciences Tripos Part IB
If z were to cross the branch cut, from above to below say, then θ 1 would be unchanged (at π) but θ 2 would “jump” from 0 to 2π. This is, of course, not allowed, as we may not cross branch cuts. So z cannot wind round just one of the branch points.
But it is now possible for z to wind around both of the branch points together. Consider a curve C which does so. Starting from the point of C on the positive real axis (where θ 1 = θ 2 = 0) and moving anti-clockwise, both θ 1 and θ 2 increase. When we have made one complete revolution and returned to the positive real axis, having encircled both branch points exactly once, θ 1 and θ 2 both suddenly “jump” from 2π back to 0. But this jump does not result in a jump in the value of θ 1 − θ 2 ; so f (z) is not affected, and is indeed single-valued as claimed.
Exactly the same choice of branch cuts occurs for the function
g(z) = (z^2 − 1) (^1) / 2 .
With the appropriate definitions of θ 1 and θ 2 , as above, the single-valued choice is
g(z) = (z − 1) (^1) / 2 (z + 1) (^1) / 2 =
r 1 r 2 ei(θ^1 +θ^2 )/^2.
This time the single branch cut works because, when both θ 1 and θ 2 jump by 2π, 1 2 (θ^1 +^ θ^2 ) jumps by 2π^ also; and^ e
2 πi (^) = 1. The cut prevents either θ 1 or θ 2 jumping on
its own.
This idea can be extended to higher numbers of branch points in the right circum- stances.
Mathematical Methods II Natural Sciences Tripos Part IB
