Complex Analysis 31 | Application of the Identity Theorem [dark version]

TL;DR

The identity theorem guarantees uniqueness: two holomorphic functions on a connected open set that agree on a set with an accumulation point in the domain must be identical everywhere.

Briefing Cornell Notes

Briefing

A holomorphic extension of a real function is essentially forced to be unique once it matches on any set with an accumulation point inside a connected complex domain. That uniqueness comes straight from the identity theorem: if two holomorphic functions agree on a set that has a limit point in the domain, they must agree everywhere on that domain. The practical payoff is that many familiar real functions—especially those defined by power series—have no “alternative” complex versions.

The example centers on cosine. Cosine is first treated as a real function on ℝ, expressed by its power series

cos(x)=∑_{k=0}^∞ (-1)^k x^{2k}/(2k)!

This series defines a C^∞ function on ℝ. Next, consider a holomorphic function G on some open connected set D ⊂ ℂ that intersects the real line (so D∩ℝ is non-empty). If G agrees with cos(x) on D∩ℝ, then the identity theorem applies because D∩ℝ contains accumulation points in D. The conclusion is rigid: G must equal the same power series representation on D. In particular, extending cosine from ℝ to the whole complex plane produces a unique holomorphic function, with no freedom to choose a different analytic continuation.

The discussion then generalizes beyond cosine. Start with a C^∞ function f on ℝ and choose an open connected complex domain D that meets ℝ. If there exists a holomorphic function G on D whose restriction to D∩ℝ matches f, then that holomorphic extension is unique. In other words, there is at most one holomorphic function on D that can extend f while agreeing with it on the real slice.

A key corollary follows for functions given by power series on ℝ: the same power series automatically defines a holomorphic function on ℂ, and the identity theorem guarantees there cannot be another holomorphic extension that still matches the original real function on the real line. This turns analytic continuation from a “maybe there are many extensions” problem into a “if an extension exists, it’s determined” principle.

With that uniqueness principle established, the chapter closes and the next step points toward the residue theorem—an essential tool for computing complicated complex integrals. The uniqueness results here set the stage: once a holomorphic extension is pinned down, later calculations can rely on a single, well-defined analytic object rather than competing candidates.

Cornell Notes

The identity theorem implies a strong uniqueness rule for holomorphic extensions. If D is a connected open set in ℂ and two holomorphic functions agree on D∩ℝ (or any set with an accumulation point in D), then the functions must be identical on all of D. Using cosine as the model, the real power series for cos(x) forces any holomorphic function G on D that matches cosine on D∩ℝ to equal the same power series on D. This extends to a general C^∞ function f on ℝ: if a holomorphic extension G exists on D, it is unique. For power-series-defined real functions, the complex holomorphic extension is therefore determined automatically by the same series.

What exact condition lets the identity theorem force two holomorphic functions to be equal everywhere?

Take a connected open set D ⊂ ℂ and two holomorphic functions f and g on D. Consider the set of points in D where f(z)=g(z). If that coincidence set has an accumulation point in D, then f and g must be identical on all of D (so f=g on D). The accumulation point requirement is what turns “agreement on many points” into “agreement everywhere” for holomorphic functions.

Why does matching cosine on D∩ℝ determine the holomorphic function on all of D?

Cosine is written on ℝ by its power series with only even powers: cos(x)=∑_{k=0}^∞ (-1)^k x^{2k}/(2k)!. Suppose G is holomorphic on an open connected D and G agrees with cos on D∩ℝ. Because D is open and connected, the real slice D∩ℝ is non-empty and has accumulation points in D. That means f(z)=cos(z) and g(z)=G(z) coincide on a set with a limit point inside D, so the identity theorem forces G to equal the same power-series-defined function on all of D.

What is the “no choice” conclusion for extending cosine to the complex plane?

Once a holomorphic extension exists that matches cosine on the real line, the identity theorem leaves no alternative. The extension must be the one given by the cosine power series. In particular, extending cosine from ℝ to the whole complex plane ℂ yields a unique holomorphic function, because any other holomorphic candidate agreeing on ℝ would have to coincide everywhere.

How does the argument generalize from cosine to an arbitrary real C^∞ function f?

Let f be C^∞ on ℝ, and let D be an open connected subset of ℂ with D∩ℝ ≠ ∅. If there is a holomorphic function G on D such that G restricted to D∩ℝ equals f restricted to D∩ℝ, then that G is unique. So there is at most one holomorphic extension matching f on the real slice; any two such extensions would agree on a set with an accumulation point in D and therefore be identical.

Why do power series on ℝ automatically give the only possible holomorphic extension?

If f on ℝ is given by a power series, the same series defines a holomorphic function on ℂ (or on the relevant complex domain of convergence). Since this holomorphic function matches f on ℝ, the identity theorem guarantees no other holomorphic function on D can match f on D∩ℝ. The power series extension is therefore the unique holomorphic extension.

Review Questions

In the identity theorem, why does an accumulation point in D matter more than just having infinitely many points of agreement?
For a holomorphic extension G on D to be unique, what role does the intersection D∩ℝ play in the argument?
How does the cosine example illustrate the relationship between real power series and holomorphic functions on ℂ?

Key Points

1
The identity theorem guarantees uniqueness: two holomorphic functions on a connected open set that agree on a set with an accumulation point in the domain must be identical everywhere.
2
To extend a real function f to a holomorphic function on a complex domain D, it’s crucial that D intersects the real line so the matching set has accumulation points in D.
3
Cosine’s real power series (with coefficients (-1)^k/(2k)! and only even powers) forces any holomorphic extension matching cosine on D∩ℝ to equal the same power-series function on D.
4
Any holomorphic extension of a C^∞ function f on ℝ (when it exists) is unique on a connected open complex domain D that meets ℝ.
5
Real functions defined by power series have a canonical holomorphic extension obtained by using the same series; no competing holomorphic extension can match on the real slice.
6
The uniqueness conclusion is “at most one extension,” not a guarantee that an extension exists for every C^∞ function—existence is separate from uniqueness.

Highlights

Matching on D∩ℝ is enough: once that set has accumulation points in D, holomorphic agreement becomes global agreement.

Cosine’s complex extension is forced by its power series—there’s no alternative holomorphic function that still matches on the real line.

For any C^∞ function on ℝ, a holomorphic extension on a connected complex domain D (if it exists) is unique.

Power-series-defined real functions extend holomorphically in a way that the identity theorem makes unambiguous.

Topics

Identity Theorem
Holomorphic Extension
Cosine Power Series
Uniqueness
Analytic Continuation