Questions: Complex Logarithm and Branch Cuts

Question 1

A student uses the principal branch to compute Log(-1) = iπ. A classmate says the answer could also be -iπ, or 3iπ, or iπ + 2πki for any integer k. Which student is right, and why?

• A. The first student — Log(-1) = iπ is the unique correct answer because the principal branch is the definition of the logarithm
• B. The classmate — all values iπ + 2πki for any integer k are valid logarithms of -1; Log gives only the principal branch value by a conventional choice, not because the others are wrong
• C. The classmate — the logarithm of a negative number is undefined, so neither answer is valid
• D. Both — iπ and -iπ are both principal branch values depending on which branch cut convention you use

Answer: (B)

The complex logarithm is genuinely multi-valued: since e^(iπ) = e^(-iπ) = e^(iπ + 2πki) = -1 for all integers k, all these are valid solutions to e^z = -1. The principal branch Log selects one of them — the one whose imaginary part lies in (-π, π] — by convention. This is a useful choice for computation but it does not make the other values 'wrong.' Multi-valuedness is intrinsic to the logarithm; the principal branch is a disambiguation tool, not the definition.

Question 2

Why must any branch of the complex logarithm have a branch cut — a curve from 0 to ∞ along which the function is discontinuous?

• A. Because the complex logarithm is not holomorphic anywhere, and a branch cut marks the region where it fails to be differentiable
• B. Because log|z| (the real part of the logarithm) is undefined at the origin, and the branch cut extends this singularity to infinity
• C. Because any continuous path around the origin forces the argument of z to increase by 2π, making a globally continuous single-valued logarithm on ℂ\{0} impossible — the discontinuity must live somewhere
• D. Because the principal argument function Arg(z) is only defined for real z, requiring a cut to extend it to the complex plane

Answer: (C)

The topological reason is definitive: if you trace a closed loop around the origin, the argument of z increases by 2π — a fact built into the geometry of the complex plane. Any single-valued continuous function claiming to be a logarithm must therefore be discontinuous at some point along that loop. The branch cut is where we accept that discontinuity. No clever formula can avoid it; it is a consequence of the plane's topology around the origin. The location of the cut is a choice; the existence of some cut is not.

Question 3

Different branch cuts for the complex logarithm are equally valid mathematically — choosing the negative real axis as a branch cut rather than the positive imaginary axis is a convention, not a mathematical necessity.

• T. True
• F. False

Answer: True

Any ray from the origin to infinity (or any curve connecting origin to ∞) can serve as a branch cut. The standard choice — the negative real axis, giving the principal branch with argument in (-π, π] — is conventional and widely used, but a branch cut along the positive imaginary axis, or any other ray, gives an equally valid single-valued holomorphic function on a different domain. What is not conventional is the need for *some* cut: that is topologically forced. The choice of *where* to place it is genuinely arbitrary.

Question 4

The multi-valuedness of the complex logarithm is a problem with the standard definition that could be resolved by choosing a better formula — one that is single-valued everywhere on ℂ\{0}.

• T. True
• F. False

Answer: False

Multi-valuedness is not a defect in the definition — it is an intrinsic consequence of e^z being periodic with period 2πi. Because e^z is not injective, its inverse must be multi-valued; no formula can change this. Any attempt to define a single-valued logarithm on all of ℂ\{0} must introduce a discontinuity somewhere (the branch cut), because a continuous loop around the origin forces the argument to change by 2π. The branch cut does not solve multi-valuedness — it relocates and acknowledges the unavoidable discontinuity.

Question 5

Explain why the complex logarithm is multi-valued, starting from what you know about the complex exponential. Why is a branch cut necessary rather than optional?

Model answer: The complex exponential e^z is periodic: e^(z + 2πi) = e^z for all z, meaning infinitely many distinct inputs map to the same output. The logarithm, as the inverse of e^z, must assign to each w all values z such that e^z = w — and there are infinitely many, each differing by a multiple of 2πi. This is multi-valuedness. To use the logarithm in analysis (integration, complex powers, etc.), we need a single-valued function, which requires choosing one value for each w. But no such choice can be made continuously everywhere on ℂ\{0}: traversing a loop around the origin forces the argument to increase by 2π, so any single-valued selection must be discontinuous somewhere. A branch cut is the curve where we place that forced discontinuity — making log single-valued and holomorphic on the cut domain, at the cost of being undefined on the cut itself.

The logical chain is: e^z periodic → log multi-valued → any single-valued version must have a discontinuity → branch cut places that discontinuity on a chosen curve. The branch cut is necessary because the complex plane has non-trivial topology around the origin: loops around it are not contractible to points, so they cannot be made argument-continuous. This topological fact forces the cut; only its location is a choice.