Logarithms and branch choices

Try

A point just above the negative real axis has principal angle close to π. A point just below it has principal angle close to −π. The points can be arbitrarily close while those angle values differ by almost a full turn. This is a branch jump caused by an angle convention, not a discontinuity of the underlying points.

For z≠0, every logarithm value has the form log|z|+iθ, where θ is any argument of z. Choosing one value independently at each point does not automatically produce a continuous function.

Predict

Predict what happens to a continuously tracked logarithm while a point makes one counterclockwise circuit around zero. Its real part returns to its starting value, but its imaginary part increases by 2π. A globally single-valued continuous logarithm would have to return to precisely the same value.

We will distinguish the multivalued inverse relation from a holomorphic branch defined on a specified open domain.

Experiment

Use the principal argument range (−π,π).

Point	Principal logarithm
1	0
i	iπ/2
−i	−iπ/2
exp(3πi/4)	3πi/4

The negative real axis and zero are excluded. Multiply the last point by itself and compare the principal logarithm of the product with twice its principal logarithm. The discrepancy is an exact 2πi.

The live experiment below has its own radius and travel-angle settings, independent of the fixed table. Compare the same endpoint after zero turns and one full turn. Track a continued logarithm along the path separately from the principal branch evaluated only at the endpoint.

Live experiment: one endpoint, different continued values

Start at the positive real number r and follow z(t)=r·exp(it) from t=0 to the selected angle θ. Compare the logarithm continued along that path with the principal branch on the plane cut along the nonpositive real axis. The endpoint alone does not record how many turns you took.

Radius r 00.512Travel angle θ -360°-315°-270°-225°-180°-135°-90°-45°0°45°90°135°180°225°270°315°360°Reset experiment

Endpoint z=(-0.707107)+i(0.707107). Travel angle θ=2.356194 radians.

Continued logarithm: 0+i(2.356194). Its exponential equals the endpoint. It starts at log(r) and tracks the travel angle continuously.

Principal logarithm: 0+i(2.356194). Its argument is restricted to (−π,π). Crossing the cut makes the principal value jump; the path continuation can remain continuous.

The continued square root, starting at positive √r, is 0.382683+i(0.92388). Squaring gives z. After one full turn it has the opposite sign from its initial value.

Along this nonzero circular path, ∫dz/z=iθ=i(2.356194). A complete turn returns to the same endpoint but has integral ±2πi, which rules out a global primitive of 1/z on the punctured plane.

Compare 0° and 360°, then 135° and 225°. Explain why a continuous path lift is not a single-valued holomorphic logarithm on the entire punctured plane. At 180°, distinguish “no value for this branch” from “no logarithm exists.” Settings are temporary; the diagrams have fixed axes and do not animate.

Understand

On the slit plane obtained by removing the nonpositive real axis, define Log z=log|z|+i Arg z with Arg z in (−π,π). Locally the angle varies continuously and its real partial derivatives satisfy the Cauchy–Riemann equations; equivalently, local inversion of the nonvanishing derivative of exp gives holomorphy. A direct coordinate calculation away from the cut gives derivatives of log|z| equal to x/(x²+y²), y/(x²+y²), and of Arg z equal to −y/(x²+y²), x/(x²+y²). These continuous partials verify the equations.

Differentiating exp(Log z)=z gives z(Log z)′=1, hence derivative 1/z. This calculation applies to every holomorphic logarithm branch, not only the principal one.

If L₁ and L₂ are two continuous logarithms on a connected domain, their difference takes values in 2πi times the integers. Continuity into that discrete set makes the difference constant. The connectedness hypothesis matters: separate components can use different integer shifts.

There is no continuous logarithm on the entire punctured plane. Along exp(it), a continuous choice differs from it by a fixed integer multiple of 2πi. At t=0 and t=2π the input is identical but these values differ by 2πi, a contradiction. This is a topological obstruction, not a failure of local differentiation.

Apply

Define a square-root branch by S(z)=exp(Log z/2) on the slit plane. Check S²=z, then differentiate to get S′=1/(2S). Explain why the denominator cannot vanish on that domain.

Repair the unrestricted logarithm product law by adding the missing integer multiple of 2πi. The principal branch need not make that integer zero even when z, w and zw all belong to its domain. A correct calculation must check their three chosen arguments separately.

Continue

Carry the domain whenever writing Log or a noninteger power zᵅ=exp(α Log z). Later, winding numbers formalize the obstruction to a global logarithm, and analytic continuation tracks branches along paths. Optional depth: construct a different branch by cutting along the positive real axis and using arguments in (0,2π).