Complex Logarithm Branches

For any nonzero complex number z = r · exp(iθ) with positive modulus r > 0, the complex logarithm is multivalued because angles differing by integer multiples of 2π map to the same point in the complex plane. Separating the single-valued real part ln(r) from the phase yields three distinct concepts:

Principal logarithm: Log(z) = ln(r) + i · Arg(z), where Arg(z) ∈ (-π, π] and on the negative real axis Arg(-r) = +π (upper-side boundary convention).
Selected branch k: log_k(z) = ln(r) + i · (Arg(z) + 2πk) for integer sheet index k ∈ {-1, 0, 1}.
Analytic continuation along a path: ln(r) + iθ, tracking the continuous accumulated angle θ from θ = 0 without jumping across the branch cut.

Worked Example: Positive Imaginary Axis vs. Winding Past the Cut

Compare the principal value, branch sheet k, and path-continued phase in two configurations:

Positive imaginary point (r = 2, θ/π = 1/2, k = 0):
z = 2 · exp(iπ/2) = 0 + 2i, so Point real = 0 and Point imaginary = 2. The log real part is ln(2) ≈ 0.6931471805599453, Arg(z)/π = 1/2 = 0.5, selected branch imaginary over π is 1/2 + 2(0) = 0.5, continued imaginary over π is 0.5, and continuation shift over π is 0.5 - 0.5 = 0.
Winding past the branch cut (r = 1, θ/π = 3/2, k = -1):
At θ = 3π/2, the point is z = -i (real 0, imaginary -1, ln(1) = 0). Its principal argument wraps back into (-π, π] as Arg(-i)/π = -1/2 = -0.5. For sheet k = -1, the selected branch imaginary over π is -1/2 + 2(-1) = -5/2 = -2.5, whereas the path-continued phase remains θ/π = 3/2 = 1.5, giving continuation shift 1.5 - (-0.5) = 2 (one full 2π winding).

Interactive Experiment

Inspect branch jumps and continuous winding in the Complex Logarithm Branches interactive workspace:

Set Radius r = 2, Angle theta over pi = 1/2, and Branch k = 0. Verify Point real = 0, Point imaginary = 2, Log real part = 0.6931471805599453, and zero continuation shift.
Set Radius r = 1, Angle theta over pi = 3/2, and Branch k = -1. Observe that Principal argument over pi = -0.5 and Continued imaginary over pi = 1.5 differ by Continuation shift over pi = 2.
Set Radius r = 1, Angle theta over pi = -1, and Branch k = 0. At z = -1 on the negative real axis, the upper-side convention sets Principal argument over pi = 1 while the clockwise path has Continued imaginary over pi = -1 (shift -2).
Compare Angle theta over pi = 2 with Branch k = 0 (one full positive winding returning to z = 1 with shift 2) against Angle theta over pi = 0 with Branch k = -1 (selected branch -2, shift 0).

Limitations and Assumptions

The complex logarithm is undefined at the branch point z = 0, so the radius r must remain strictly positive. Assigning Arg(-r) = +π on the negative real axis is a boundary convention that makes Log(z) continuous from the upper half-plane Im(z) > 0, but it does not make the principal branch analytic across the cut: approaching the negative real axis from Im(z) < 0 yields limit -π, a jump of 2πi.

References

The principal value of the complex logarithm, its branch cut along the negative real axis, and Riemann-surface sheet continuations are defined in the following reference:

NIST Digital Library of Mathematical Functions (DLMF), Section 4.2(i): Complex Logarithm Definitions and Principal Value

Complex Logarithm Branches