Branch Cuts and Multivalued Functions — Mathematics Knowledge System

Overview

Functions like the complex logarithm and the square root are not single-valued: every nonzero z has infinitely many logarithms (differing by multiples of 2πi) and every nonzero z has two square roots. To turn such a rule into an honest analytic function, you must choose a branch — a consistent, continuous selection of one value at each point on a restricted domain. The obstruction to doing this on all of ℂ\{0} is topological: walking a small loop around the origin changes arg(z) by 2π, so log(z) = ln|z| + i·arg(z) cannot return to its starting value after a full loop unless you forbid such loops. A branch cut is a curve (often a ray or segment) removed from the domain specifically to block these loops, making arg(z) — and hence log(z) and z^(1/2) — single-valued and holomorphic on what remains. The endpoint(s) of the cut, where the multivaluedness is unavoidable no matter how you cut, are called branch points.

Intuition

Think of arg(z) as an odometer reading the angle swept out as you walk around the origin. Start at z=1 with arg=0 and walk counterclockwise all the way around a circle back to z=1: the odometer now reads 2π, not 0, even though you're standing on the same point. Any function built from arg — log(z), z^(1/2), z^α for non-integer α — inherits this discontinuity: it 'wants' to jump by 2πi (or a fraction of it) every time you circle the origin. A branch cut is a wall you build (conventionally along the negative real axis) that you're simply not allowed to cross, so no path in the cut domain can ever wind around the origin, and the odometer becomes single-valued and continuous — at the cost of a jump discontinuity across the wall itself. The branch point at the center is where the wall must start, because it's the one place no cut can remove the winding.

Formal Definition

Definition

The principal branch of the logarithm uses the principal argument Arg(z) ∈ (−π, π], cutting along the negative real axis (including 0):

\operatorname{Log}(z) = \ln|z| + i\,\operatorname{Arg}(z), \qquad \operatorname{Arg}(z) \in (-\pi, \pi]

Principal branch of the logarithm

z^{1/2} = \sqrt{|z|}\, e^{i\operatorname{Arg}(z)/2}, \qquad \operatorname{Arg}(z) \in (-\pi, \pi]

Principal branch of the square root (same cut)

\operatorname{Log}(z) \text{ is holomorphic on } \mathbb{C} \setminus (-\infty, 0], \qquad \frac{d}{dz}\operatorname{Log}(z) = \frac{1}{z}

Domain of holomorphy and derivative

z = 0 \text{ (and } \infty \text{) is a branch point of } \operatorname{Log} \text{ and } z^{1/2}

Branch points

Worked Examples

For z=-1: |z|=1 so ln|z|=0, and Arg(-1)=π (the principal argument in (-π,π] for a negative real number).
$\operatorname{Log}(-1) = \ln 1 + i\pi = i\pi$
For z=i: |z|=1 so ln|z|=0, and Arg(i)=π/2 (i lies on the positive imaginary axis).
$\operatorname{Log}(i) = \ln 1 + i\frac{\pi}{2} = \frac{i\pi}{2}$

Answer: Log(-1) = iπ and Log(i) = iπ/2.

Practice Problems

Difficulty 5/10

Using the principal branch, compute Log(1) and Log(-i).

Difficulty 6/10

Using the principal branch of the square root, compute (-4)^(1/2).

Difficulty 8/10

Explain why Log(z) cannot be extended to a continuous function on all of ℂ\{0}, using a loop around the origin at radius 1.

Quiz

Summary

Multivalued expressions like log(z) and z^(1/2) arise because arg(z) changes by 2π after a loop around the origin.
A branch is a continuous, single-valued selection of the multivalued expression on a restricted domain; a branch cut is the curve removed to block encircling loops.
The principal branch uses Arg(z) ∈ (−π, π] and cuts along the negative real axis; Log(-1) = iπ and (4i)^(1/2) = √2 + i√2 under this convention.
Branch points (like z=0 for log and square root) are where no choice of cut can remove the multivaluedness — some cut must terminate there.

References

WebsiteWikipedia — Branch point
WebsiteWikipedia — Complex logarithm