File: Mathlib/Analysis/Complex/RemovableSingularity.lean

full_name: Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
code: /-- **Removable singularity** theorem, weak version. If `f : ℂ → E` is differentiable in a punctured
neighborhood of a point and is continuous at this point, then it is analytic at this point. -/
theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c) : AnalyticAt ℂ f c 
goal: AnalyticAt ℂ f c

full_name: Complex.differentiableOn_compl_singleton_and_continuousAt_iff
code: theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s : Set ℂ} {c : ℂ}
    (hs : s ∈ 𝓝 c) :
    DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s 
goal: DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s

full_name: Complex.differentiableOn_dslope
code: theorem differentiableOn_dslope {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c) :
    DifferentiableOn ℂ (dslope f c) s ↔ DifferentiableOn ℂ f s 
goal: DifferentiableOn ℂ (dslope f c) s ↔ DifferentiableOn ℂ f s

full_name: Complex.differentiableOn_update_limUnder_of_isLittleO
code: /-- **Removable singularity** theorem: if `s` is a neighborhood of `c : ℂ`, a function `f : ℂ → E`
is complex differentiable on `s \ {c}`, and $f(z) - f(c)=o((z-c)^{-1})$, then `f` redefined to be
equal to `limUnder (𝓝[≠] c) f` at `c` is complex differentiable on `s`. -/
theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
    (hd : DifferentiableOn ℂ f (s \ {c}))
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) s 
goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) s

full_name: Complex.differentiableOn_update_limUnder_insert_of_isLittleO
code: /-- **Removable singularity** theorem: if `s` is a punctured neighborhood of `c : ℂ`, a function
`f : ℂ → E` is complex differentiable on `s`, and $f(z) - f(c)=o((z-c)^{-1})$, then `f` redefined to
be equal to `limUnder (𝓝[≠] c) f` at `c` is complex differentiable on `{c} ∪ s`. -/
theorem differentiableOn_update_limUnder_insert_of_isLittleO {f : ℂ → E} {s : Set ℂ} {c : ℂ}
    (hc : s ∈ 𝓝[≠] c) (hd : DifferentiableOn ℂ f s)
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) (insert c s) 
goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) (insert c s)

full_name: Complex.differentiableOn_update_limUnder_of_bddAbove
code: /-- **Removable singularity** theorem: if `s` is a neighborhood of `c : ℂ`, a function `f : ℂ → E`
is complex differentiable and is bounded on `s \ {c}`, then `f` redefined to be equal to
`limUnder (𝓝[≠] c) f` at `c` is complex differentiable on `s`. -/
theorem differentiableOn_update_limUnder_of_bddAbove {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
    (hd : DifferentiableOn ℂ f (s \ {c})) (hb : BddAbove (norm ∘ f '' (s \ {c}))) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) s 
goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) s

full_name: Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO
code: /-- **Removable singularity** theorem: if a function `f : ℂ → E` is complex differentiable on a
punctured neighborhood of `c` and $f(z) - f(c)=o((z-c)^{-1})$, then `f` has a limit at `c`. -/
theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z)
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) 
goal: Filter.Tendsto f (𝓝[≠] c) (𝓝 (limUnder (𝓝[≠] c) f))

full_name: Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under
code: /-- **Removable singularity** theorem: if a function `f : ℂ → E` is complex differentiable and
bounded on a punctured neighborhood of `c`, then `f` has a limit at `c`. -/
theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z)
    (hb : IsBoundedUnder (· ≤ ·) (𝓝[≠] c) fun z => ‖f z - f c‖) :
    Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) 
goal: Filter.Tendsto f (𝓝[≠] c) (𝓝 (limUnder (𝓝[≠] c) f))

full_name: Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
code: /-- The Cauchy formula for the derivative of a holomorphic function. -/
theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U : Set ℂ}
    (hU : IsOpen U) {c w₀ : ℂ} {R : ℝ} {f : ℂ → E} (hc : closedBall c R ⊆ U)
    (hf : DifferentiableOn ℂ f U) (hw₀ : w₀ ∈ ball c R) :
    ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀ 
goal: ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀