Uniqueness principle for analytic functions #

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

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theorem analytic_on.eq_on_zero_of_preconnected_of_eventually_eq_zero_aux {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_add_comm_group F] [normed_space 𝕜 F] [complete_space F] {f : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hU : is_preconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[nhds z₀] 0) :

set.eq_on f 0 U

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 Superseded by eq_on_zero_of_preconnected_of_locally_zero which does not assume completeness of the target space.

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theorem analytic_on.eq_on_zero_of_preconnected_of_eventually_eq_zero {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_add_comm_group F] [normed_space 𝕜 F] {f : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hU : is_preconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[nhds z₀] 0) :

set.eq_on 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 eq_on_zero_of_preconnected_of_frequently_eq_zero.

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theorem analytic_on.eq_on_of_preconnected_of_eventually_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_add_comm_group F] [normed_space 𝕜 F] {f g : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hg : analytic_on 𝕜 g U) (hU : is_preconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[nhds z₀] g) :

set.eq_on 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 eq_on_of_preconnected_of_frequently_eq.

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theorem analytic_on.eq_of_eventually_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_add_comm_group F] [normed_space 𝕜 F] {f g : E → F} [preconnected_space E] (hf : analytic_on 𝕜 f set.univ) (hg : analytic_on 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.