Lecture 10: The Special Cauchy's Formula and Applications (Text 118–126)

This file formalizes Morera's theorem as stated in the textbook (Ahlfors, Theorem 11, p. 122), which is the main result from Lecture 10 that is used throughout the rest of the course (particularly in the proof of the General Cauchy Theorem, Lecture 13).

Main results

• morera_disk: Morera's theorem for a disk. If f is continuous on a disk and its rectangle integrals vanish (i.e., f is conservative), then f has a primitive on the disk.

• morera_disk_differentiableOn: Same as above, but concluding DifferentiableOn ℂ f (ball c r).

• morera_theorem: Morera's theorem (Theorem 11, p. 122). If f is continuous on an open set Ω ⊆ ℂ and the rectangle integrals of f vanish throughout Ω, then f is holomorphic on Ω.

• morera_theorem_iff: Morera's theorem — iff version. On an open set, a function is holomorphic if and only if it is continuous with vanishing rectangle integrals. This combines Morera's theorem (⟸) with the Cauchy–Goursat theorem (⟹).

Mathematical context

Morera's theorem is the converse of Cauchy's theorem for rectangles. The textbook states it as:

Theorem 11 (Morera). If f(z) is defined and continuous in a region Ω, and if ∫_γ f(z) dz = 0 for every closed curve γ in Ω, then f(z) is analytic in Ω.

The proof proceeds as follows: the vanishing integral condition implies that f has a local primitive F in every disk contained in Ω. Since F is analytic (having a complex derivative f), and the derivative of an analytic function is analytic, f itself is analytic.

In Mathlib's language, the "vanishing rectangle integrals" condition is captured by Complex.IsConservativeOn f Ω, which says that for all rectangles R ⊆ Ω, the wedge integral satisfies wedgeIntegral z w f = -wedgeIntegral w z f. The existence of a primitive is Complex.IsExactOn f Ω, meaning ∃ g, ∀ z ∈ Ω, HasDerivAt g (f z) z.

References

• Ahlfors, Complex Analysis, 3rd ed., Theorem 11 (p. 122)
• Lecture notes, Lecture 10 (Text 118–126)
• Used in Lecture 13 (General Cauchy Theorem) to show holomorphicity of h

theorem morera_disk {f : ℂ → ℂ} {c : ℂ} {r : ℝ} (hcont : ContinuousOn f (Metric.ball c r)) (hcons : Complex.IsConservativeOn f (Metric.ball c r)) : Complex.IsExactOn f (Metric.ball c r)

Morera's theorem for a disk (Text, p. 122). If f is continuous on a disk and its rectangle integrals vanish, then f has a primitive (i.e., there exists F with F' = f) on the disk.

In the textbook's language: a continuous function with vanishing contour integrals has an antiderivative. Since the antiderivative is holomorphic and the derivative of a holomorphic function is holomorphic, f itself is holomorphic.

theorem morera_disk_differentiableOn {f : ℂ → ℂ} {c : ℂ} {r : ℝ} (hcont : ContinuousOn f (Metric.ball c r)) (hcons : Complex.IsConservativeOn f (Metric.ball c r)) : DifferentiableOn ℂ f (Metric.ball c r)

Morera's theorem for a disk — differentiability version. Under the same hypotheses as morera_disk, f is holomorphic on the disk.

theorem morera_theorem {f : ℂ → ℂ} {Ω : Set ℂ} (hΩ : IsOpen Ω) (hcont : ContinuousOn f Ω) (hcons : Complex.IsConservativeOn f Ω) : DifferentiableOn ℂ f Ω

Morera's theorem (Theorem 11, p. 122). If f is continuous on an open set Ω ⊆ ℂ and the rectangle integrals of f vanish throughout Ω, then f is holomorphic on Ω.

Formally: IsConservativeOn f Ω ∧ ContinuousOn f Ω → DifferentiableOn ℂ f Ω.

The textbook states this as: if f is continuous in a region Ω and ∫_γ f(z) dz = 0 for every closed curve γ in Ω, then f is analytic in Ω.

The proof localizes to disks: for each z ∈ Ω, pick a ball B(z, r) ⊆ Ω. The hypotheses restrict to the ball, so morera_disk gives a primitive on the ball. Having a primitive implies differentiability at z.

theorem morera_theorem_iff {f : ℂ → ℂ} {Ω : Set ℂ} (hΩ : IsOpen Ω) : Complex.IsConservativeOn f Ω ∧ ContinuousOn f Ω ↔ DifferentiableOn ℂ f Ω

Morera's theorem — iff version (Text, p. 122). On an open set, a function is holomorphic if and only if it is continuous with vanishing rectangle integrals. This combines Morera's theorem (⟸) with the Cauchy–Goursat theorem (⟹).

theorem cauchy_goursat_conservative {f : ℂ → ℂ} {Ω : Set ℂ} (hΩ : IsOpen Ω) (hf : DifferentiableOn ℂ f Ω) : Complex.IsConservativeOn f Ω ∧ ContinuousOn f Ω

Cauchy–Goursat direction of Morera's theorem. If f is holomorphic on an open set Ω, then f is continuous on Ω and its rectangle integrals vanish throughout Ω. This is the "easy" direction.