An analytic formula can outgrow the power series that first introduced it—but only along paths where its analytic structure can be continued consistently.
Complex analysis turns local information into remarkably rigid global information. A holomorphic function is locally represented by a power series. The Identity Theorem makes continuation unique whenever overlapping pieces genuinely agree. But logarithms and roots reveal a second fact: local analytic definitions can fail to glue into one global single-valued function because loops around excluded points may return a different value.
This guide owns analytic continuation, branches, branch cuts, branch points and introductory monodromy for R26. It uses the contour and topology machinery already developed on BTT while leaving conformal mapping, special functions, several complex variables and broader applications to R26.05–R26.08.
Prerequisites: Complex Functions and the Complex Plane, Holomorphic Functions and Complex Differentiation, and Contour Integration and Residues.
Reading route: local analytic data → overlapping disks → Identity Theorem → continuation along paths → uniqueness → geometric-series example → singularity barriers → complex logarithm → branches and cuts → roots and powers → winding obstruction → monodromy → square-root sign change → simply connected domains → branch points → Riemann surfaces → natural boundaries → verification workflow → practice → solutions.
1. Analytic continuation starts from local analytic data
Suppose f is holomorphic on an open set U and g is holomorphic on a larger or different open set V.
If U∩V contains a nonempty open region on which f=g, then g is an analytic continuation of f from U into V.
The phrase does not mean “extend the formula by wishful substitution”. It means construct another holomorphic object that agrees where both definitions are valid.
2. The Identity Theorem makes continuation unique
If two holomorphic functions on a connected domain agree on a set having an accumulation point inside the domain, then they agree everywhere on that domain.
Therefore two competing analytic continuations that overlap and agree on enough local data cannot later diverge while remaining holomorphic on one connected overlap.
Uniqueness is one of the main reasons analytic continuation is meaningful.
3. A germ records a function only near one point
Two holomorphic functions defined near a point a determine the same germ at a if they agree on some neighbourhood of a.
A germ forgets how the function was written far away and remembers only its local analytic behaviour.
Analytic continuation can be viewed as transporting this local germ through overlapping neighbourhoods.
4. Overlapping disks carry a local power series forward
Suppose a Taylor series centred at a converges in a disk D(a,R). Pick b inside that disk. The resulting analytic function has a Taylor series centred at b with its own radius of convergence.
If the new disk reaches beyond the first, the b-centred series extends the same analytic function farther.
Repeating this process creates a chain of overlapping analytic neighbourhoods.
5. A Taylor radius stops at analytic obstruction, not at arbitrary notation
For a holomorphic function represented around a, the Taylor series converges at least until the nearest singular obstruction of the analytic function in the complex plane.
For rational and many standard meromorphic functions, the distance to the nearest pole determines the Taylor radius.
The original power series may stop even though the analytic function itself continues by another local expansion.
6. Worked continuation: the geometric series
For |z|<1,
1+z+z²+⋯=1/(1−z).
The power series diverges for |z|>1, but the rational function 1/(1−z) is holomorphic throughout C\{1}.
Thus 1/(1−z) is the analytic continuation of the geometric-series function beyond its original unit disk, except across its actual pole at z=1.
7. The radius of one series is not the domain of the function
A frequent mistake is to identify a function with one chosen Taylor expansion.
The expansion around a point is a local representation. The analytic function can often be represented by different Taylor series in different overlapping regions.
The function is the analytic object; the series is one coordinate chart for it.
8. Continuation along a path uses a chain of neighbourhoods
Let γ:[0,1]→Ω be a path. Start with an analytic germ near γ(0). Cover the path by overlapping disks and continue the germ from one disk to the next.
If the process succeeds all the way to γ(1), we say the germ has been analytically continued along γ.
The continuation can depend on the path when the domain has nontrivial loops around branch points.
9. Local uniqueness does not automatically imply global path independence
At each small overlap, the Identity Theorem forces a unique next germ.
But two different long paths from the same start to the same endpoint may wind differently around excluded points.
If continuation along those paths produces different endpoint germs, the phenomenon is monodromy.
10. The complex logarithm is the central branch example
Because e^{w}=z means
w=ln|z|+i(arg z+2πk), k∈Z,
the inverse of the exponential is naturally multivalued on C\{0}.
There is no single-valued holomorphic logarithm on the entire punctured plane.
11. A branch chooses one analytic value consistently on a domain
A branch of log on a domain D⊂C\{0} is a holomorphic function L:D→C satisfying
e^{L(z)}=z.
Then differentiating gives
L'(z)=1/z.
The branch encodes a continuous choice of argument on D.
12. The principal logarithm uses a slit domain
A standard principal branch is defined on
C\(−∞,0]
by choosing Arg z∈(−π,π) and setting
Log z=ln|z|+i Arg z.
The negative real axis is removed so the chosen argument does not jump by 2π inside the domain.
13. A branch cut is a domain choice, not an intrinsic line of singular points
The logarithm has an intrinsic branch point at 0, but the negative real axis in the principal branch is a chosen cut.
One could instead remove another ray from 0 to infinity and choose a different continuous interval of arguments.
Different cuts support different branches of the same local analytic object.
14. Integrating 1/z explains the obstruction to a global logarithm
If a holomorphic logarithm L existed on all of C\{0}, then L’=1/z.
Every closed contour integral of L’ would be zero because derivatives have primitives.
But around the unit circle, ∮dz/z=2πi.
Contradiction. The winding detected by contour integration is exactly the global branch obstruction.
15. Simply connected domains avoiding zero admit logarithm branches
If D is simply connected, D⊂C\{0}, then 1/z has a primitive on D.
That primitive can be exponentiated and normalised to construct a holomorphic logarithm branch.
Thus topology and analysis meet directly: removing nontrivial winding makes the inverse of exp single-valued.
16. Square roots are logarithm branches divided by two
On a domain carrying a logarithm branch L, define
√z = exp(L(z)/2).
Then (√z)²=z on that domain.
Different logarithm branches can produce different root branches.
17. nth roots have n local sheets
Similarly,
z^{1/n}=exp(L(z)/n)
defines one branch on a domain with a chosen log branch.
Globally around 0, the n possible values are related by multiplication by nth roots of unity.
18. General complex powers require a logarithm branch
For α∈C, define on a chosen branch domain
z^α=exp(αL(z)).
If α is an integer, the result is branch-independent. If α is noninteger, changing L by 2πik can change the value.
So familiar exponent laws must be used with branch conditions visible.
19. Worked branch law failure
With principal logarithms, the identity
Log(zw)=Log z+Log w
can fail because principal arguments are forced back into (−π,π).
For z=w=−1, principal Log(−1)=iπ, so the right side is 2πi, whereas principal Log(1)=0.
The exponential of both sides agrees, but the selected branches differ by 2πi.
20. Monodromy records what continuation does around loops
Begin with one local value of log z near z=1 and analytically continue it once counterclockwise around the origin.
After returning to the starting point, the continued value has increased by
2πi.
The point is the same, but the analytic branch has moved to another sheet.
21. A square root changes sign after one loop around zero
Locally write √z=e^{(1/2)Log z}.
After one positive circuit around zero, Log z gains 2πi, so
e^{(1/2)(Log z+2πi)}=−e^{(1/2)Log z}.
After two circuits the original value returns. The square root therefore has two sheets around its branch point.
22. Branch points are not isolated poles
A pole or essential singularity is an isolated point around which the function is single-valued on a punctured neighbourhood.
A branch point behaves differently: analytic continuation around a loop can move the function to another value.
For log z and z^{1/n}, zero is a branch point. Infinity may also carry branch structure when the Riemann sphere is considered.
23. Branch cuts make a multivalued relation single-valued on a selected sheet
Cutting the plane along a curve joining a branch point to another branch point or to infinity can remove problematic loops.
On the resulting domain, one chooses a continuous argument and thereby a single-valued analytic branch.
The cut is bookkeeping for the chosen sheet; it should not be confused with an ordinary line of poles.
24. The Monodromy Theorem gives path independence under stronger topology
A standard form says that if an analytic germ can be continued along every path in a simply connected domain and continuation is locally well-defined, then continuation from a fixed start to a fixed endpoint is independent of path.
Informally: if every loop can be continuously shrunk without crossing an obstruction, continuation cannot acquire nontrivial monodromy.
Precise versions state technical hypotheses carefully; simple connectedness is the geometric core.
25. Covering spaces explain logarithm monodromy geometrically
The exponential map exp:C→C\{0} is a covering map.
A path in C\{0} can be lifted to C once a starting logarithm value is chosen. A loop winding k times around zero lifts to a path whose endpoint differs by 2πik.
This is the topology behind the analytic statement that logarithm continuation changes by 2πik.
26. Riemann surfaces replace multivalued functions with single-valued functions on a larger space
Rather than cutting the plane and choosing one branch, one can build a surface whose different sheets record the possible values.
On the Riemann surface of √z, two sheets are glued so that moving around the branch point transfers one sheet to the other.
On the logarithm surface, infinitely many sheets record argument changes by multiples of 2π.
The formerly multivalued relation becomes a single-valued holomorphic function on its natural surface.
27. Analytic continuation cannot cross a genuine singularity
The geometric-series continuation 1/(1−z) cannot be made holomorphic through z=1 because z=1 is an actual pole.
By contrast, the circle |z|=1 away from z=1 is not itself a singular barrier for that function; it was only the convergence boundary of the original series.
Always distinguish a representation boundary from an analytic obstruction.
28. Singularities determine local continuation geometry
Poles, essential singularities and branch points obstruct continuation in different ways.
A meromorphic continuation may legitimately pass through a former domain boundary while introducing isolated poles. A branch continuation may require a covering surface rather than a single plane domain.
The word “continue” therefore does not promise that the result remains entire or single-valued on the original plane.
29. A natural boundary is stronger than one isolated obstruction
Some power series cannot be analytically continued across any point of their circle of convergence. That circle is then a natural boundary.
Lacunary series provide classical examples under appropriate gap conditions.
This is very different from the geometric series, whose unit-circle convergence boundary contains only one genuine singularity of its continuation.
30. Analytic continuation of special functions is a later R26 job
The Gamma function, zeta function and related special functions are often first defined by series or integrals in restricted half-planes, then analytically or meromorphically continued.
R26.06 owns those special-function constructions and asymptotic methods.
This article supplies the continuation logic without pre-empting that later owner.
31. Analytic continuation is not numerical extrapolation
Fitting a curve outside observed data and analytically continuing a holomorphic function are fundamentally different operations.
Analytic continuation is constrained by exact local holomorphic agreement and uniqueness theorems. Numerical extrapolation is an inference procedure whose reliability depends on a model and data.
The similar everyday word “continuation” should not blur that distinction.
32. Common theorem-boundary errors
- A Taylor series’ convergence circle is not automatically a singular boundary of the analytic function.
- The Identity Theorem requires an accumulation point inside the connected domain, not merely infinitely many agreement points accumulating only at the boundary.
- A branch cut is chosen; a branch point is intrinsic.
- There is no global holomorphic logarithm on C\{0}.
- Principal Log does not obey Log(zw)=Log z+Log w without branch qualifications.
- Noninteger complex powers depend on the chosen logarithm branch.
- Local uniqueness of continuation does not by itself guarantee path independence around nontrivial loops.
- Poles, essential singularities and branch points are different analytic phenomena.
33. A dependable branch-and-continuation workflow
Start by stating the initial domain and local analytic definition. Identify the true singularities and excluded points before extending anything.
When using logarithms, arguments, roots or noninteger powers, state the branch domain and argument interval explicitly.
For continuation along paths, ask whether different paths can wind around an obstruction. If yes, test monodromy rather than assuming path independence.
Use contour integrals such as ∮dz/z or covering-space lifts to detect global obstruction. Treat a chosen cut as a modelling device for one sheet, not as proof that the entire removed curve is singular.
34. Independent practice: twenty questions
- Define analytic continuation from U to V using overlap.
- Why is an analytic continuation unique on a connected overlap?
- What is a germ of a holomorphic function?
- Explain how overlapping Taylor disks can continue a function.
- Why does the geometric series stop at |z|=1 while 1/(1−z) continues farther?
- What is continuation along a path?
- Write all logarithms of z≠0.
- Define a branch of the logarithm.
- State a standard principal Log domain and argument range.
- Why is the negative real axis in principal Log called a branch cut rather than a line of poles?
- Why can no holomorphic logarithm exist on C\{0}?
- On a simply connected domain avoiding zero, why can a logarithm branch exist?
- Define a square-root branch using a log branch.
- What happens to log z after one positive circuit around zero?
- What happens to √z after one positive circuit around zero?
- Distinguish a branch point from a pole.
- State the geometric idea of the Monodromy Theorem.
- How does exp:C→C\{0} explain logarithm monodromy?
- What is a Riemann surface used for in this context?
- What is a natural boundary?
35. Worked solutions and checks
1. A holomorphic g on V continues f on U when f and g agree on a nonempty connected overlap containing enough local data; commonly they agree on an open overlap.
2. The Identity Theorem forces two holomorphic candidates agreeing on a set with an interior accumulation point to agree throughout the connected overlap.
3. An equivalence class of holomorphic functions that agree on some neighbourhood of the chosen point.
4. Re-expand at a point inside the current convergence disk; the new Taylor series may reach beyond the old disk while agreeing on their overlap.
5. The original series has radius 1 because of the singularity at z=1, but its sum equals the rational function 1/(1−z), which is holomorphic everywhere else.
6. Transport a starting analytic germ through a chain of overlapping neighbourhoods covering the path.
7. ln|z|+i(arg z+2πk), k∈Z.
8. A single-valued holomorphic L on a chosen domain satisfying e^{L(z)}=z.
9. C\(−∞,0] with Arg z∈(−π,π) is a standard choice.
10. Another ray could be removed instead; the cut prevents discontinuous argument wrapping but points on the ray are not intrinsically all poles.
11. A global log would have derivative 1/z, forcing ∮dz/z=0 around every closed contour, contradicting the value 2πi around the unit circle.
12. Simple connectedness removes the winding obstruction, so 1/z has a primitive that can be normalised as a logarithm.
13. Choose L and define √z=exp(L(z)/2).
14. It increases by 2πi.
15. It changes sign and returns to its original value only after a second circuit.
16. A pole is an isolated single-valued singularity with finite principal part; a branch point causes continuation around loops to move between values/sheets.
17. In a simply connected region with continuation available along every path, endpoint continuation is path-independent.
18. Exp is a covering map; loops around zero lift to paths ending at values shifted by integer multiples of 2πi.
19. It enlarges the domain so a formerly multivalued relation can become one single-valued holomorphic function on an appropriate sheeted surface.
20. A boundary across which analytic continuation is impossible at every point.
36. Where Batch 01 hands off
R26.01–R26.04 now establish the foundational one-variable complex-analysis route: complex geometry and functions → holomorphic differentiation → Cauchy and residue calculus → analytic continuation, branches and monodromy.
R26.05–R26.08 remain separate future cells: conformal mapping and potential theory, special functions and asymptotics, several complex variables, and applications/verification. Their presence in the Atlas is not yet a claim of published coverage.
Sources and further study
For an undergraduate complex-variables route, see MIT OpenCourseWare 18.04 Complex Variables with Applications. For the topology of path lifting, loops and coverings, continue through BTT’s Fundamental Groups and Covering Spaces.