Casorati-Weierstrass Theorem (Theorem 9, Lecture 11)

A holomorphic function comes arbitrarily close to any complex value in every neighborhood of an essential singularity.

Main definitions

• IsEssentialSingularity: A function f : ℂ → ℂ has an essential singularity at c if it is holomorphic on a punctured neighborhood of c but is not meromorphic at c.

Main results

• casorati_weierstrass: Theorem 9, Lecture 11. If f has an essential singularity at c, then for every w : ℂ, every δ > 0, and every ε > 0, there exists z with 0 < ‖z - c‖ < δ and ‖f z - w‖ < ε.
• casorati_weierstrass_dense: Equivalent formulation: the image of any punctured ball around c under f is dense in ℂ.

Proof outline

The proof is by contradiction, following the textbook argument:

1. Suppose f misses a ball B(w₀, ε) in a punctured neighborhood of c.
2. Then g(z) = (f(z) - w₀)⁻¹ is bounded by ε⁻¹ and holomorphic near c.
3. By the removable singularity theorem, g extends to a meromorphic function at c.
4. Since f = w₀ + g⁻¹, the function f is also meromorphic at c.
5. This contradicts the hypothesis that c is an essential singularity.

def IsEssentialSingularity (f : ℂ → ℂ) (c : ℂ) : Prop

A function f : ℂ → ℂ has an essential singularity at c if f is holomorphic (complex differentiable) on a punctured neighborhood of c but is not meromorphic at c.

This corresponds to Definition (iii) in Lecture 11: an isolated singularity that is neither removable nor a pole. In Mathlib's framework, removable singularities and poles are exactly the meromorphic functions (MeromorphicAt), so an essential singularity is characterized by ¬ MeromorphicAt f c.

theorem meromorphicAt_of_differentiableAt_punctured_bounded {f : ℂ → ℂ} {c : ℂ} {M : ℝ} (hd : ∀ᶠ (z : ℂ) in nhdsWithin c {c}ᶜ, DifferentiableAt ℂ f z) (hb : ∀ᶠ (z : ℂ) in nhdsWithin c {c}ᶜ, ‖f z‖ ≤ M) : MeromorphicAt f c

If f is holomorphic on a punctured neighborhood of c and bounded there, then f is meromorphic at c. This follows from Riemann's removable singularity theorem: the bounded holomorphic function has a removable singularity and extends analytically.

theorem casorati_weierstrass {f : ℂ → ℂ} {c : ℂ} (hd : ∀ᶠ (z : ℂ) in nhdsWithin c {c}ᶜ, DifferentiableAt ℂ f z) (hnotmero : ¬MeromorphicAt f c) (w : ℂ) (δ : ℝ) (hδ : 0 < δ) (ε : ℝ) (hε : 0 < ε) : ∃ (z : ℂ), 0 < ‖z - c‖ ∧ ‖z - c‖ < δ ∧ ‖f z - w‖ < ε

Casorati-Weierstrass theorem (Theorem 9, Lecture 11): A holomorphic function comes arbitrarily close to any complex value in every neighborhood of an essential singularity.

If f is holomorphic on a punctured neighborhood of c and is not meromorphic at c (i.e., c is an essential singularity), then for every w : ℂ and every ε > 0 and δ > 0, there exists z with 0 < ‖z - c‖ < δ and ‖f z - w‖ < ε.

theorem casorati_weierstrass_dense {f : ℂ → ℂ} {c : ℂ} (hf : IsEssentialSingularity f c) (r : ℝ) (hr : 0 < r) : Dense (f '' (Metric.ball c r \ {c}))

Casorati-Weierstrass theorem, dense image formulation (Theorem 9, Lecture 11): The image of any punctured ball around an essential singularity is dense in ℂ.