Complex Analysis 31 | Application of the Identity Theorem

TL;DR

The identity theorem states that two holomorphic functions on a connected open set D that agree on a set with an accumulation point in D must be identical on all of D.

Briefing Cornell Notes

Briefing

The identity theorem doesn’t just prove two holomorphic functions must match—it also forces uniqueness when extending real functions into the complex plane. If two holomorphic functions agree on a set with an accumulation point inside a connected open domain, they must be identical everywhere on that domain. That principle becomes a practical tool: once a real function is known on the real line and a holomorphic extension exists on a complex domain that touches the real axis, there is no freedom left in how the extension can look.

A concrete example uses the cosine function. Cosine is first treated as a real function defined on ℝ via its standard power series, with coefficients (−1)^k/(2k)! and only even powers of x. This series defines a C∞ function on ℝ. Now consider any holomorphic function G defined on an open connected set D ⊂ ℂ that intersects the real line (so D ∩ ℝ is non-empty). If G agrees with cosine on D ∩ ℝ, then the identity theorem applies because D ∩ ℝ contains accumulation points in D. The result is decisive: G must coincide with the same power series representation throughout D. In particular, extending cosine to the entire complex plane forces the unique holomorphic extension given by the usual cosine power series—there is no alternative holomorphic continuation consistent with the real-axis values.

The same logic generalizes beyond cosine. Start with any C∞ function f on ℝ, and choose a connected open set D in ℂ that intersects ℝ. If there exists a holomorphic function G on D whose restriction to D ∩ ℝ equals f’s restriction there, then G is unique. In other words, any holomorphic extension of f from the real line into such a complex domain is determined completely by the values on the real-axis slice.

This uniqueness matters because many important real functions are defined by power series. When f is given by a power series on ℝ, the corresponding power series automatically defines a holomorphic function on ℂ, and the identity theorem guarantees that any holomorphic extension matching f on the real line must be exactly that one. The takeaway is less about computing new functions and more about eliminating ambiguity: once a holomorphic extension exists under the identity theorem’s conditions, its form is fixed.

With this chapter closed, the focus shifts toward the residue theorem, a tool designed to compute difficult complex integrals—an application area where having rigid analytic structure from results like the identity theorem becomes especially valuable.

Cornell Notes

The identity theorem implies a strong uniqueness principle for holomorphic extensions. On a connected open set D ⊂ ℂ, if two holomorphic functions agree on a subset with an accumulation point in D, then they agree everywhere on D. Applying this to cosine, the real cosine function defined by its even-power series has a unique holomorphic extension: any holomorphic G on D that matches cosine on D ∩ ℝ must equal the same power series on all of D. More generally, if f is C∞ on ℝ and D is connected open with D ∩ ℝ ≠ ∅, then any holomorphic extension G of f to D is unique. For power-series-defined functions on ℝ, the complex power series provides that unique holomorphic continuation.

What exact condition forces two holomorphic functions to be identical on a connected open set?

Let D be a connected open domain in ℂ. If f and g are holomorphic on D and the set {z ∈ D : f(z) = g(z)} has an accumulation point in D, then f = g on all of D. Agreement at infinitely many points isn’t enough by itself—the key is that those agreement points accumulate inside D.

Why does cosine have no “alternative” holomorphic extension once it matches on the real axis?

Cosine is defined on ℝ by its power series with only even powers: cos(x) = Σ_{k≥0} (−1)^k x^{2k}/(2k)!. Take a connected open set D ⊂ ℂ that intersects ℝ, and a holomorphic function G on D such that G(x) = cos(x) for x ∈ D ∩ ℝ. Because D is open, D ∩ ℝ has accumulation points in D. The identity theorem then forces G to equal the same power series representation throughout D, leaving no other holomorphic continuation consistent with the real-axis values.

What role does the intersection D ∩ ℝ ≠ ∅ play in the extension argument?

The extension problem requires a “starting agreement set” where the complex function can be compared to the real function. If D ∩ ℝ were empty, there would be no real-axis points inside D on which G could be required to match f. With D ∩ ℝ non-empty and D open, that intersection also provides accumulation points in D, which is what activates the identity theorem.

How does the identity theorem generalize the cosine example to an arbitrary C∞ function on ℝ?

Given f ∈ C∞(ℝ) and a connected open set D ⊂ ℂ with D ∩ ℝ ≠ ∅, there can be at most one holomorphic function G on D such that G restricted to D ∩ ℝ equals f restricted to D ∩ ℝ. Any holomorphic extension that matches f on the real slice is forced to be unique by the identity theorem.

Why do power series on ℝ automatically determine the holomorphic extension on ℂ?

If f on ℝ is given by a power series, the same power series defines a holomorphic function on ℂ (within its radius of convergence, and in the cosine case it converges everywhere). That power-series-defined holomorphic function is one candidate extension. The identity theorem then guarantees it is the only holomorphic extension that agrees with f on D ∩ ℝ, so the power series fixes the complex continuation uniquely.

Review Questions

In the identity theorem, why is an accumulation point inside D essential rather than just infinitely many agreement points?
For a connected open set D that intersects ℝ, what does the identity theorem say about the number of holomorphic extensions of a given C∞ function f on ℝ?
How does the cosine power series (even powers with coefficients (−1)^k/(2k)!) lead to uniqueness of its holomorphic extension?

Key Points

1
The identity theorem states that two holomorphic functions on a connected open set D that agree on a set with an accumulation point in D must be identical on all of D.
2
To extend a real function into ℂ, the complex domain D must intersect the real line so there is a real-axis agreement set.
3
For cosine, any holomorphic function G on D that matches cosine on D ∩ ℝ must equal the cosine power series on all of D.
4
A C∞ function f on ℝ has at most one holomorphic extension to any connected open set D ⊂ ℂ with D ∩ ℝ ≠ ∅.
5
When a real function is defined by a power series, the corresponding complex power series gives the unique holomorphic continuation consistent with the real-axis values.

Highlights

Cosine’s holomorphic extension is forced: any holomorphic G matching cosine on D ∩ ℝ must equal the same even-power series everywhere on D.

Uniqueness hinges on accumulation points inside the connected open domain, not just agreement at isolated points.

For any C∞ function on ℝ, holomorphic continuation into a connected open complex domain touching ℝ is unique if it exists.

Power series defined on ℝ determine their complex holomorphic extensions with no ambiguity under the identity theorem.

Topics

Identity Theorem
Holomorphic Extension
Cosine Power Series
Uniqueness
Residue Theorem