Two holomorphic functions f and g are defined on the same connected open domain D. A student discovers that f(1/n) = g(1/n) for every positive integer n. What can the Identity Theorem conclude?
Af and g agree on all rational inputs in D, but may differ on irrational inputs
Bf and g are identical throughout all of D, because the sequence {1/n} has a limit point (0) inside D
Cf and g agree on an interval around 0, but may diverge further from the origin
DNo conclusion can be drawn without knowing the Taylor series of both functions at every point
The Identity Theorem requires only that two holomorphic functions agree on a set with a limit point inside the connected domain — and {1/n} converges to 0, a limit point. That single condition forces f ≡ g throughout all of D. This rigidity has no analogue in real analysis: a smooth real function could be modified arbitrarily on any interval while agreeing with another function on {1/n}. The remarkable power of complex analyticity is that values on any limit-point-containing set determine the function globally.
Question 2 Multiple Choice
A mathematician continues log(z) starting from z = 1 (where log(1) = 0) along a path that winds once counterclockwise around the origin and returns to z = 1. What value does the continuation assign to z = 1 after this loop?
A0 — the continuation returns to the starting value because z = 1 is the same point
B2πi — the continuation tracks the accumulated argument, which increased by 2π around the origin
CThe continuation is undefined because log is not holomorphic along a circular path
D−2πi — the continuation loses one branch worth of argument when returning to the start
This is monodromy in action. As z travels counterclockwise around the origin, its argument increases by 2π. Since log(z) = ln|z| + i·arg(z), the imaginary part accumulates 2π over one full loop, returning log(1) with value 0 + 2πi = 2πi, not 0. Analytic continuation is locally unique — no other extension exists in any overlapping disk — but globally path-dependent around branch points. This path-dependence is what a Riemann surface resolves by separating the branches into distinct sheets.
Question 3 True / False
The Identity Theorem implies that a holomorphic function is more rigidly determined by local data than any smooth real-valued function, because matching values on a set with a limit point forces global equality on the entire connected domain.
TTrue
FFalse
Answer: True
The Explainer explicitly contrasts this with real analysis: 'A smooth real function can be freely modified on any interval without affecting its values elsewhere. A holomorphic complex function has no such freedom.' The Identity Theorem's hypothesis requires only a set with a limit point (not even density), yet the conclusion is global identity throughout the connected domain. This rigidity is the source of analytic continuation's power — it means that any holomorphic extension to an overlapping domain is unique.
Question 4 True / False
Analytic continuation usually produces the same value regardless of the path taken, because the Identity Theorem guarantees that holomorphic extensions are unique.
TTrue
FFalse
Answer: False
The Identity Theorem guarantees local uniqueness — there is only one holomorphic extension in any overlapping disk. But global continuation along paths can be path-dependent when the path encircles branch points (like the origin for log z or z^(1/2)). Monodromy — returning a different value after a closed loop — is a consequence of the multi-valued nature of these functions in the plane. Path-independence holds only in simply connected domains containing no branch points; the Riemann surface resolves the ambiguity by providing a globally single-valued domain.
Question 5 Short Answer
The Riemann zeta function is defined by Σ n⁻ˢ, which converges only for Re(s) > 1. Why does analytic continuation matter for understanding ζ(s) beyond this region?
Think about your answer, then reveal below.
Model answer: Analytic continuation uniquely extends ζ(s) to all of ℂ minus the pole at s = 1, because the Identity Theorem guarantees that any two holomorphic functions agreeing on an open set must be identical throughout their connected domain. The extended function is the unique analytic function that matches the series where the series converges. This matters because the extended function reveals properties invisible in the convergent region — including the zeros in the critical strip 0 < Re(s) < 1, whose distribution is the subject of the Riemann Hypothesis.
This example shows why analytic continuation is not merely a technical extension trick but a conceptual expansion of what a function is. The series Σ n⁻ˢ is not 'the' Riemann zeta function — it is a representation valid in one region. The analytically continued function, which has no series representation in the critical strip, is the complete mathematical object, and understanding the distinction is essential for complex analysis and analytic number theory.