# Documentation

Mathlib.Analysis.Complex.RemovableSingularity

# Removable singularity theorem #

In this file we prove Riemann's removable singularity theorem: if f : ℂ → E is complex differentiable in a punctured neighborhood of a point c and is bounded in a punctured neighborhood of c (or, more generally, $f(z) - f(c)=o((z-c)^{-1})$), then it has a limit at c and the function update f c (limUnder (𝓝[≠] c) f) is complex differentiable in a neighborhood of c.

theorem Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {E : Type u} [] [] {f : E} {c : } (hd : ∀ᶠ (z : ) in nhdsWithin c {c}, ) (hc : ) :

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 Complex.differentiableOn_compl_singleton_and_continuousAt_iff {E : Type u} [] [] {f : E} {s : } {c : } (hs : s nhds c) :
theorem Complex.differentiableOn_dslope {E : Type u} [] [] {f : E} {s : } {c : } (hc : s nhds c) :
theorem Complex.differentiableOn_update_limUnder_of_isLittleO {E : Type u} [] [] {f : E} {s : } {c : } (hc : s nhds c) (hd : DifferentiableOn f (s \ {c})) (ho : (fun z => f z - f c) =o[nhdsWithin c {c}] fun z => (z - c)⁻¹) :

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 Complex.differentiableOn_update_limUnder_insert_of_isLittleO {E : Type u} [] [] {f : E} {s : } {c : } (hc : s nhdsWithin c {c}) (hd : ) (ho : (fun z => f z - f c) =o[nhdsWithin c {c}] fun z => (z - c)⁻¹) :

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 Complex.differentiableOn_update_limUnder_of_bddAbove {E : Type u} [] [] {f : E} {s : } {c : } (hc : s nhds c) (hd : DifferentiableOn f (s \ {c})) (hb : BddAbove (norm f '' (s \ {c}))) :

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 Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {E : Type u} [] [] {f : E} {c : } (hd : ∀ᶠ (z : ) in nhdsWithin c {c}, ) (ho : (fun z => f z - f c) =o[nhdsWithin c {c}] fun z => (z - c)⁻¹) :

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 Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {E : Type u} [] [] {f : E} {c : } (hd : ∀ᶠ (z : ) in nhdsWithin c {c}, ) (hb : Filter.IsBoundedUnder (fun x x_1 => x x_1) (nhdsWithin c {c}) fun z => f z - f c) :

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 Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {E : Type u} [] [] {U : } (hU : ) {c : } {w₀ : } {R : } {f : E} (hc : U) (hf : ) (hw₀ : w₀ ) :
(()⁻¹ ∮ (z : ) in C(c, R), ((z - w₀) ^ 2)⁻¹ f z) = deriv f w₀

The Cauchy formula for the derivative of a holomorphic function.