A local logarithm near one has value zero there. Follow it continuously around the unit circle, and its value becomes it along exp(it). After one full turn, the input returns to one while the continued logarithm has value 2πi.
Analytic continuation follows a local function through overlapping neighborhoods. It is not the same as repeatedly resetting a principal branch. Resetting would introduce a jump, whereas continuation preserves the local analytic agreement along the path.
Predict
Predict the corresponding effect for a square-root germ that starts with value one. Along the same circle it becomes exp(it/2), ending at minus one. After two turns it returns to one.
Predict whether every germ can continue across every point of the plane. The germ 1/(1−z) near zero encounters an actual pole at one. Removing branch ambiguity cannot remove such a singularity.
Experiment
Track continuation around zero.
Function germ at 1
Start
After one turn
After two turns
Logarithm
0
2πi
4πi
Square root
1
−1
1
Reciprocal
1
1
1
The table records path-followed germs, not values of one global branch on the punctured plane. The reciprocal is already a global holomorphic function there.
Understand
A germ records a holomorphic function near a point, identifying two representatives when they agree on some smaller neighborhood. To continue a germ along a path, cover successive path segments by overlapping disks and choose holomorphic representatives agreeing on the overlaps used. Identity uniqueness makes each continuation step unique once such an extension exists. It does not automatically supply existence at every later point.
The monodromy theorem states that if a germ at a base point admits analytic continuation along every path in a simply connected domain, these continuations determine a single-valued holomorphic function on that domain. Both hypotheses matter. Simple connectivity removes distinct loop classes, while the continuation assumption rules out inaccessible singularities.
The full finite proof is now developed in analytic elements and path stability and compact homotopies and monodromy. They construct adapted disk chains, positive segment and seam margins, finite compatible strips and the local holomorphy of the endpoint assignment. These are the ingredients that a compact-square picture alone cannot supply.
For logarithms on the punctured plane, continuation exists along every path but simple connectivity fails, and different winding numbers produce different values. On a slit plane the loop obstruction is removed, producing a single branch. A cut is a domain choice: a nonzero point on the cut still has a small neighborhood admitting a local branch.
Apply
Repair “The plane is simply connected, so the germ 1/(1−z) extends to an entire function.” The germ cannot be continued holomorphically through one. A hypothetical entire extension would agree with the rational function away from one by the identity theorem and contradict its unbounded behavior near the pole.
As a concrete action, draw two paths from one to i, one with an extra circuit around zero. Track their logarithm values and explain the difference 2πi without introducing a discontinuity along either individual path.
Continue
This completes the sequence from local differentiability to global contour and branch reasoning. Continue with the linked finite monodromy proof. Further study includes Riemann surfaces, the Riemann mapping theorem and Picard theory. For independent work, write a contour or mapping report that states its domain, proves the necessary estimates, and separates numerical observations from theorem-based conclusions.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Start the square-root germ with value 1 at z=1 and continue it twice counterclockwise around zero. What is its final value at 1?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A standard monodromy theorem assumes a germ can be continued:
Explain why analytic continuation through a chain of overlapping disks is unique once agreement holds on each overlap component used by the chain.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Track the logarithm germ with value zero at 1 once around γ(t)=exp(it), 0≤t≤2π.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show a square-root germ changes sign after one turn around zero and explain why a branch cut does not create a singularity at every cut point.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does simple connectivity alone not guarantee continuation of an arbitrary germ to an entire chosen domain? Give an example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
