File: Mathlib/Analysis/Complex/Liouville.lean

full_name: «_aux_Mathlib_Analysis_Complex_Liouville___macroRules_term_̂_1»
code: local postfix:100 "̂" => UniformSpace.Completion
goal: Lean.Macro

full_name: _aux_Mathlib_Analysis_Complex_Liouville___unexpand_UniformSpace_Completion_1
code: local postfix:100 "̂" => UniformSpace.Completion
goal: Lean.PrettyPrinter.Unexpander

full_name: Complex.deriv_eq_smul_circleIntegral
code: /-- If `f` is complex differentiable on an open disc with center `c` and radius `R > 0` and is
continuous on its closure, then `f' c` can be represented as an integral over the corresponding
circle.

TODO: add a version for `w ∈ Metric.ball c R`.

TODO: add a version for higher derivatives. -/
theorem deriv_eq_smul_circleIntegral [CompleteSpace F] {R : ℝ} {c : ℂ} {f : ℂ → F} (hR : 0 < R)
    (hf : DiffContOnCl ℂ f (ball c R)) :
    deriv f c = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z 
goal: deriv f c = (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c) ^ (-2) • f z

full_name: Complex.norm_deriv_le_aux
code: theorem norm_deriv_le_aux [CompleteSpace F] {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
    (hf : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖deriv f c‖ ≤ C / R 
goal: ‖deriv f c‖ ≤ C / R

full_name: Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
code: /-- If `f` is complex differentiable on an open disc of radius `R > 0`, is continuous on its
closure, and its values on the boundary circle of this disc are bounded from above by `C`, then the
norm of its derivative at the center is at most `C / R`. -/
theorem norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
    (hd : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖deriv f c‖ ≤ C / R 
goal: ‖deriv f c‖ ≤ C / R

full_name: Complex.liouville_theorem_aux
code: /-- An auxiliary lemma for Liouville's theorem `Differentiable.apply_eq_apply_of_bounded`. -/
theorem liouville_theorem_aux {f : ℂ → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
    (z w : ℂ) : f z = f w 
goal: f z = f w

full_name: Differentiable.apply_eq_apply_of_bounded
code: /-- **Liouville's theorem**: a complex differentiable bounded function `f : E → F` is a constant. -/
theorem apply_eq_apply_of_bounded {f : E → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
    (z w : E) : f z = f w 
goal: f z = f w

full_name: Differentiable.exists_const_forall_eq_of_bounded
code: /-- **Liouville's theorem**: a complex differentiable bounded function is a constant. -/
theorem exists_const_forall_eq_of_bounded {f : E → F} (hf : Differentiable ℂ f)
    (hb : IsBounded (range f)) : ∃ c, ∀ z, f z = c 
goal: ∃ c, ∀ (z : E), f z = c

full_name: Differentiable.exists_eq_const_of_bounded
code: /-- **Liouville's theorem**: a complex differentiable bounded function is a constant. -/
theorem exists_eq_const_of_bounded {f : E → F} (hf : Differentiable ℂ f)
    (hb : IsBounded (range f)) : ∃ c, f = const E c 
goal: ∃ c, f = Function.const E c

full_name: Differentiable.eq_const_of_tendsto_cocompact
code: /-- A corollary of Liouville's theorem where the function tends to a finite value at infinity
(i.e., along `Filter.cocompact`, which in proper spaces coincides with `Bornology.cobounded`). -/
theorem eq_const_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
    (hb : Tendsto f (cocompact E) (𝓝 c)) : f = Function.const E c 
goal: f = Function.const E c

full_name: Differentiable.apply_eq_of_tendsto_cocompact
code: /-- A corollary of Liouville's theorem where the function tends to a finite value at infinity
(i.e., along `Filter.cocompact`, which in proper spaces coincides with `Bornology.cobounded`). -/
theorem apply_eq_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
    (x : E) (hb : Tendsto f (cocompact E) (𝓝 c)) : f x = c 
goal: f x = c