Principle of isolated zeros

This file proves the fact that the zeros of a non-constant analytic function of one variable are isolated. It also introduces a little bit of API in the HasFPowerSeriesAt namespace that is useful in this setup.

Main results

• AnalyticAt.eventually_eq_zero_or_eventually_ne_zero is the main statement that if a function is analytic at z₀, then either it is identically zero in a neighborhood of z₀, or it does not vanish in a punctured neighborhood of z₀.
• AnalyticOn.eqOn_of_preconnected_of_frequently_eq is the identity theorem for analytic functions: if a function f is analytic on a connected set U and is zero on a set with an accumulation point in U then f is identically 0 on U.

theorem HasSum.hasSum_at_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (a : ℕ → E) : HasSum (fun n => 0 ^ n • a n) (a 0)

theorem HasSum.exists_hasSum_smul_of_apply_eq_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : E} {n : ℕ} {z : 𝕜} {a : ℕ → E} (hs : HasSum (fun m => z ^ m • a m) s) (ha : ∀ (k : ℕ), k < n → a k = 0) : ∃ t, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t

theorem HasFPowerSeriesAt.has_fpower_series_dslope_fslope {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) : HasFPowerSeriesAt (dslope f z₀) (FormalMultilinearSeries.fslope p) z₀

theorem HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (n : ℕ) (hp : HasFPowerSeriesAt f p z₀) : HasFPowerSeriesAt ((Function.swap dslope z₀)^[n] f) (FormalMultilinearSeries.fslope^[n] p) z₀

theorem HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : (𝕜 → E)^[Function.swap dslope z₀] (FormalMultilinearSeries.order p) f z₀ ≠ 0

theorem HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) : ∀ᶠ (z : 𝕜) in nhds z₀, f z = (z - z₀) ^ FormalMultilinearSeries.order p • (𝕜 → E)^[Function.swap dslope z₀] (FormalMultilinearSeries.order p) f z

theorem HasFPowerSeriesAt.locally_ne_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ 0

theorem HasFPowerSeriesAt.locally_zero_iff {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ (z : 𝕜) in nhds z₀, f z = 0) ↔ p = 0

theorem AnalyticAt.eventually_eq_zero_or_eventually_ne_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) : (∀ᶠ (z : 𝕜) in nhds z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ 0

The principle of isolated zeros for an analytic function, local version: if a function is analytic at z₀, then either it is identically zero in a neighborhood of z₀, or it does not vanish in a punctured neighborhood of z₀.

theorem AnalyticAt.eventually_eq_or_eventually_ne {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {g : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) : (∀ᶠ (z : 𝕜) in nhds z₀, f z = g z) ∨ ∀ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ g z

theorem AnalyticAt.frequently_zero_iff_eventually_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w) : (∃ᶠ (z : 𝕜) in nhdsWithin w {w}ᶜ, f z = 0) ↔ ∀ᶠ (z : 𝕜) in nhds w, f z = 0

theorem AnalyticAt.frequently_eq_iff_eventually_eq {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {g : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) : (∃ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z = g z) ↔ ∀ᶠ (z : 𝕜) in nhds z₀, f z = g z

theorem AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfw : ∃ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z = 0) : Set.EqOn f 0 U

The principle of isolated zeros for an analytic function, global version: if a function is analytic on a connected set U and vanishes in arbitrary neighborhoods of a point z₀ ∈ U, then it is identically zero in U. For higher-dimensional versions requiring that the function vanishes in a neighborhood of z₀, see AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero.

theorem AnalyticOn.eqOn_zero_of_preconnected_of_mem_closure {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) : Set.EqOn f 0 U

theorem AnalyticOn.eqOn_of_preconnected_of_frequently_eq {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {g : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z = g z) : Set.EqOn f g U

The identity principle for analytic functions, global version: if two functions are analytic on a connected set U and coincide at points which accumulate to a point z₀ ∈ U, then they coincide globally in U. For higher-dimensional versions requiring that the functions coincide in a neighborhood of z₀, see AnalyticOn.eqOn_of_preconnected_of_eventuallyEq.

theorem AnalyticOn.eqOn_of_preconnected_of_mem_closure {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {g : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) : Set.EqOn f g U

theorem AnalyticOn.eq_of_frequently_eq {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {g : 𝕜 → E} {z₀ : 𝕜} [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f Set.univ) (hg : AnalyticOn 𝕜 g Set.univ) (hfg : ∃ᶠ (z : 𝕜) in nhdsWithin z₀ {z₀}ᶜ, f z = g z) : f = g

The identity principle for analytic functions, global version: if two functions on a normed field 𝕜 are analytic everywhere and coincide at points which accumulate to a point z₀, then they coincide globally. For higher-dimensional versions requiring that the functions coincide in a neighborhood of z₀, see AnalyticOn.eq_of_eventuallyEq.