# Documentation

Mathlib.Analysis.Analytic.Uniqueness

# Uniqueness principle for analytic functions #

We show that two analytic functions which coincide around a point coincide on whole connected sets, in AnalyticOn.eqOn_of_preconnected_of_eventuallyEq.

theorem AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux {𝕜 : Type u_1} {E : Type u_2} [] {F : Type u_3} [] [] {f : EF} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : ) {z₀ : E} (h₀ : z₀ U) (hfz₀ : f =ᶠ[nhds z₀] 0) :
Set.EqOn f 0 U

If an analytic function vanishes around a point, then it is uniformly zero along a connected set. Superseded by eqOn_zero_of_preconnected_of_locally_zero which does not assume completeness of the target space.

theorem AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero {𝕜 : Type u_1} {E : Type u_2} [] {F : Type u_3} [] {f : EF} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : ) {z₀ : E} (h₀ : z₀ U) (hfz₀ : f =ᶠ[nhds z₀] 0) :
Set.EqOn f 0 U

The identity principle for analytic functions: If an analytic function vanishes in a whole neighborhood of a point z₀, then it is uniformly zero along a connected set. For a one-dimensional version assuming only that the function vanishes at some points arbitrarily close to z₀, see eqOn_zero_of_preconnected_of_frequently_eq_zero.

theorem AnalyticOn.eqOn_of_preconnected_of_eventuallyEq {𝕜 : Type u_1} {E : Type u_2} [] {F : Type u_3} [] {f : EF} {g : EF} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : ) {z₀ : E} (h₀ : z₀ U) (hfg : f =ᶠ[nhds z₀] g) :
Set.EqOn f g U

The identity principle for analytic functions: If two analytic functions coincide in a whole neighborhood of a point z₀, then they coincide globally along a connected set. For a one-dimensional version assuming only that the functions coincide at some points arbitrarily close to z₀, see eqOn_of_preconnected_of_frequently_eq.

theorem AnalyticOn.eq_of_eventuallyEq {𝕜 : Type u_1} {E : Type u_2} [] {F : Type u_3} [] {f : EF} {g : EF} (hf : AnalyticOn 𝕜 f Set.univ) (hg : AnalyticOn 𝕜 g Set.univ) {z₀ : E} (hfg : f =ᶠ[nhds z₀] g) :
f = g

The identity principle for analytic functions: If two analytic functions on a normed space coincide in a neighborhood of a point z₀, then they coincide everywhere. For a one-dimensional version assuming only that the functions coincide at some points arbitrarily close to z₀, see eq_of_frequently_eq.