analysis.complex.removable_singularity
⟷
Mathlib.Analysis.Complex.RemovableSingularity
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-import Analysis.Calculus.FderivAnalytic
+import Analysis.Calculus.FDeriv.Analytic
import Analysis.Asymptotics.SpecificAsymptotics
import Analysis.Complex.CauchyIntegral
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -91,7 +91,7 @@ theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set
suffices DifferentiableOn ℂ F (s \ {c}) ∧ ContinuousAt F c
by
rw [differentiable_on_compl_singleton_and_continuous_at_iff hc, ← differentiable_on_dslope hc,
- dslope_sub_smul] at this <;>
+ dslope_sub_smul] at this <;>
try infer_instance
have hc : tendsto f (𝓝[≠] c) (𝓝 (deriv F c)) :=
continuous_at_update_same.mp (this.continuous_on.continuous_at hc)
@@ -143,7 +143,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f :
(ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) :=
by
- rw [eventually_nhdsWithin_iff] at hd
+ rw [eventually_nhdsWithin_iff] at hd
have : DifferentiableOn ℂ f ({z | z ≠ c → DifferentiableAt ℂ f z} \ {c}) := fun z hz =>
(hz.1 hz.2).DifferentiableWithinAt
have H := differentiable_on_update_lim_of_is_o hd this ho
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-import Mathbin.Analysis.Calculus.FderivAnalytic
-import Mathbin.Analysis.Asymptotics.SpecificAsymptotics
-import Mathbin.Analysis.Complex.CauchyIntegral
+import Analysis.Calculus.FderivAnalytic
+import Analysis.Asymptotics.SpecificAsymptotics
+import Analysis.Complex.CauchyIntegral
#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.complex.removable_singularity
-! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Analysis.Calculus.FderivAnalytic
import Mathbin.Analysis.Asymptotics.SpecificAsymptotics
import Mathbin.Analysis.Complex.CauchyIntegral
+#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
/-!
# Removable singularity theorem
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -35,6 +35,7 @@ variable {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace
namespace Complex
+#print Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt /-
/-- **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}
@@ -50,7 +51,9 @@ theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ
(has_fpower_series_on_ball_of_differentiable_off_countable (countable_singleton c) hc
(fun z hz => hRs (diff_subset_diff_left ball_subset_closed_ball hz)) hR0).AnalyticAt
#align complex.analytic_at_of_differentiable_on_punctured_nhds_of_continuous_at Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
+-/
+#print Complex.differentiableOn_compl_singleton_and_continuousAt_iff /-
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 :=
by
@@ -66,7 +69,9 @@ theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s
simpa only [DifferentiableWithinAt, HasFDerivWithinAt, hne.nhds_within_diff_singleton] using
hd x ⟨hx, hne⟩
#align complex.differentiable_on_compl_singleton_and_continuous_at_iff Complex.differentiableOn_compl_singleton_and_continuousAt_iff
+-/
+#print Complex.differentiableOn_dslope /-
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 :=
⟨fun h => h.of_dslope, fun h =>
@@ -74,7 +79,9 @@ theorem differentiableOn_dslope {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s
⟨Iff.mpr (differentiableOn_dslope_of_nmem fun h => h.2 rfl) (h.mono <| diff_subset _ _),
continuousAt_dslope_same.2 <| h.DifferentiableAt hc⟩⟩
#align complex.differentiable_on_dslope Complex.differentiableOn_dslope
+-/
+#print Complex.differentiableOn_update_limUnder_of_isLittleO /-
/-- **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 `lim (𝓝[≠] c) f` at `c` is complex differentiable on `s`. -/
@@ -99,7 +106,9 @@ theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set
(continuous_within_at_id.tendsto.sub tendsto_const_nhds).smul tendsto_const_nhds
simpa [← smul_add, ContinuousWithinAt] using H.add H'
#align complex.differentiable_on_update_lim_of_is_o Complex.differentiableOn_update_limUnder_of_isLittleO
+-/
+#print Complex.differentiableOn_update_limUnder_insert_of_isLittleO /-
/-- **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 `lim (𝓝[≠] c) f` at `c` is complex differentiable on `{c} ∪ s`. -/
@@ -110,7 +119,9 @@ theorem differentiableOn_update_limUnder_insert_of_isLittleO {f : ℂ → E} {s
differentiableOn_update_limUnder_of_isLittleO (insert_mem_nhds_iff.2 hc)
(hd.mono fun z hz => hz.1.resolve_left hz.2) ho
#align complex.differentiable_on_update_lim_insert_of_is_o Complex.differentiableOn_update_limUnder_insert_of_isLittleO
+-/
+#print Complex.differentiableOn_update_limUnder_of_bddAbove /-
/-- **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
`lim (𝓝[≠] c) f` at `c` is complex differentiable on `s`. -/
@@ -125,7 +136,9 @@ theorem differentiableOn_update_limUnder_of_bddAbove {f : ℂ → E} {s : Set
mem_nhdsWithin_iff_exists_mem_nhds_inter.2
⟨s, hc, fun z hz => norm_sub_le_of_le (hC <| mem_image_of_mem _ hz) le_rfl⟩⟩
#align complex.differentiable_on_update_lim_of_bdd_above Complex.differentiableOn_update_limUnder_of_bddAbove
+-/
+#print Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO /-
/-- **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}
@@ -139,7 +152,9 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f :
have H := differentiable_on_update_lim_of_is_o hd this ho
exact continuousAt_update_same.1 (H.differentiable_at hd).ContinuousAt
#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO
+-/
+#print Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under /-
/-- **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}
@@ -148,7 +163,9 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f
Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) :=
tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO hd hb.isLittleO_sub_self_inv
#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_bounded_under Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under
+-/
+#print Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable /-
/-- 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)
@@ -181,6 +198,7 @@ theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U :
simp only [dslope_of_ne, metric.sphere_disjoint_ball.ne_of_mem hz hw₀, slope, ← smul_assoc, sq,
mul_inv, Ne.def, not_false_iff, vsub_eq_sub, Algebra.id.smul_eq_mul]
#align complex.two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
+-/
end Complex
mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
! This file was ported from Lean 3 source module analysis.complex.removable_singularity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.Analysis.Complex.CauchyIntegral
/-!
# Removable singularity theorem
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
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
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3
@@ -150,7 +150,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f
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 :=
+ (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = deriv f w₀ :=
by
-- We apply the removable singularity theorem and the Cauchy formula to `dslope f w₀`
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (nonempty_of_mem hw₀).ne_empty)
@@ -170,7 +170,7 @@ theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U :
exact h1.smul (hf.continuous_on.mono (sphere_subset_closed_ball.trans hc))
have h3 : CircleIntegrable (fun z : ℂ => ((z - w₀) ^ 2)⁻¹ • f w₀) c R :=
ContinuousOn.circleIntegrable (pos_of_mem_ball hw₀).le (h1.smul continuousOn_const)
- have h4 : (∮ z : ℂ in C(c, R), ((z - w₀) ^ 2)⁻¹) = 0 := by
+ have h4 : ∮ z : ℂ in C(c, R), ((z - w₀) ^ 2)⁻¹ = 0 := by
simpa using circleIntegral.integral_sub_zpow_of_ne (by decide : (-2 : ℤ) ≠ -1) c w₀ R
simp only [smul_sub, circleIntegral.integral_sub h2 h3, h4, circleIntegral.integral_smul_const,
zero_smul, sub_zero]
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -147,7 +147,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f
#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_bounded_under Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under
/-- 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}
+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 :=
@@ -177,7 +177,7 @@ theorem two_pi_i_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U :
· refine' circleIntegral.integral_congr (pos_of_mem_ball hw₀).le fun z hz => _
simp only [dslope_of_ne, metric.sphere_disjoint_ball.ne_of_mem hz hw₀, slope, ← smul_assoc, sq,
mul_inv, Ne.def, not_false_iff, vsub_eq_sub, Algebra.id.smul_eq_mul]
-#align complex.two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable Complex.two_pi_i_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
+#align complex.two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
end Complex
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -131,7 +131,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f :
Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) :=
by
rw [eventually_nhdsWithin_iff] at hd
- have : DifferentiableOn ℂ f ({ z | z ≠ c → DifferentiableAt ℂ f z } \ {c}) := fun z hz =>
+ have : DifferentiableOn ℂ f ({z | z ≠ c → DifferentiableAt ℂ f z} \ {c}) := fun z hz =>
(hz.1 hz.2).DifferentiableWithinAt
have H := differentiable_on_update_lim_of_is_o hd this ho
exact continuousAt_update_same.1 (H.differentiable_at hd).ContinuousAt
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -42,7 +42,7 @@ theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ
replace hc : ContinuousOn f (closed_ball c R)
· refine' fun z hz => ContinuousAt.continuousWithinAt _
rcases eq_or_ne z c with (rfl | hne)
- exacts[hc, (hRs ⟨hz, hne⟩).ContinuousAt]
+ exacts [hc, (hRs ⟨hz, hne⟩).ContinuousAt]
exact
(has_fpower_series_on_ball_of_differentiable_off_countable (countable_singleton c) hc
(fun z hz => hRs (diff_subset_diff_left ball_subset_closed_ball hz)) hR0).AnalyticAt
@@ -84,7 +84,7 @@ theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set
suffices DifferentiableOn ℂ F (s \ {c}) ∧ ContinuousAt F c
by
rw [differentiable_on_compl_singleton_and_continuous_at_iff hc, ← differentiable_on_dslope hc,
- dslope_sub_smul] at this <;>
+ dslope_sub_smul] at this <;>
try infer_instance
have hc : tendsto f (𝓝[≠] c) (𝓝 (deriv F c)) :=
continuous_at_update_same.mp (this.continuous_on.continuous_at hc)
@@ -130,7 +130,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f :
(ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) :=
by
- rw [eventually_nhdsWithin_iff] at hd
+ rw [eventually_nhdsWithin_iff] at hd
have : DifferentiableOn ℂ f ({ z | z ≠ c → DifferentiableAt ℂ f z } \ {c}) := fun z hz =>
(hz.1 hz.2).DifferentiableWithinAt
have H := differentiable_on_update_lim_of_is_o hd this ho
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -24,7 +24,7 @@ function `function.update f c (lim (𝓝[≠] c) f)` is complex differentiable i
open TopologicalSpace Metric Set Filter Asymptotics Function
-open Topology Filter NNReal Real
+open scoped Topology Filter NNReal Real
universe u
mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952
@@ -60,7 +60,7 @@ theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s
refine' eventually_nhdsWithin_iff.2 ((eventually_mem_nhds.2 hs).mono fun z hz hzx => _)
exact hd.differentiable_at (inter_mem hz (is_open_ne.mem_nhds hzx))
·
- simpa only [DifferentiableWithinAt, HasFderivWithinAt, hne.nhds_within_diff_singleton] using
+ simpa only [DifferentiableWithinAt, HasFDerivWithinAt, hne.nhds_within_diff_singleton] using
hd x ⟨hx, hne⟩
#align complex.differentiable_on_compl_singleton_and_continuous_at_iff Complex.differentiableOn_compl_singleton_and_continuousAt_iff
mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65
@@ -75,7 +75,7 @@ theorem differentiableOn_dslope {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s
/-- **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 `lim (𝓝[≠] c) f` at `c` is complex differentiable on `s`. -/
-theorem differentiableOn_update_limUnder_of_isOCat {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
+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 :=
@@ -95,18 +95,18 @@ theorem differentiableOn_update_limUnder_of_isOCat {f : ℂ → E} {s : Set ℂ}
have H' : tendsto (fun z => (z - c) • f c) (𝓝[≠] c) (𝓝 (F c)) :=
(continuous_within_at_id.tendsto.sub tendsto_const_nhds).smul tendsto_const_nhds
simpa [← smul_add, ContinuousWithinAt] using H.add H'
-#align complex.differentiable_on_update_lim_of_is_o Complex.differentiableOn_update_limUnder_of_isOCat
+#align complex.differentiable_on_update_lim_of_is_o Complex.differentiableOn_update_limUnder_of_isLittleO
/-- **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 `lim (𝓝[≠] c) f` at `c` is complex differentiable on `{c} ∪ s`. -/
-theorem differentiableOn_update_limUnder_insert_of_isOCat {f : ℂ → E} {s : Set ℂ} {c : ℂ}
+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) :=
- differentiableOn_update_limUnder_of_isOCat (insert_mem_nhds_iff.2 hc)
+ differentiableOn_update_limUnder_of_isLittleO (insert_mem_nhds_iff.2 hc)
(hd.mono fun z hz => hz.1.resolve_left hz.2) ho
-#align complex.differentiable_on_update_lim_insert_of_is_o Complex.differentiableOn_update_limUnder_insert_of_isOCat
+#align complex.differentiable_on_update_lim_insert_of_is_o Complex.differentiableOn_update_limUnder_insert_of_isLittleO
/-- **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
@@ -114,8 +114,8 @@ is complex differentiable and is bounded on `s \ {c}`, then `f` redefined to be
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 :=
- differentiableOn_update_limUnder_of_isOCat hc hd <|
- IsBoundedUnder.isOCat_sub_self_inv <|
+ differentiableOn_update_limUnder_of_isLittleO hc hd <|
+ IsBoundedUnder.isLittleO_sub_self_inv <|
let ⟨C, hC⟩ := hb
⟨C + ‖f c‖,
eventually_map.2 <|
@@ -125,7 +125,7 @@ theorem differentiableOn_update_limUnder_of_bddAbove {f : ℂ → E} {s : Set
/-- **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_isOCat {f : ℂ → E} {c : 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) :=
@@ -135,7 +135,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isOCat {f : ℂ
(hz.1 hz.2).DifferentiableWithinAt
have H := differentiable_on_update_lim_of_is_o hd this ho
exact continuousAt_update_same.1 (H.differentiable_at hd).ContinuousAt
-#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isOCat
+#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO
/-- **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`. -/
@@ -143,7 +143,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f
(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) :=
- tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isOCat hd hb.isOCat_sub_self_inv
+ tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO hd hb.isLittleO_sub_self_inv
#align complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_bounded_under Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under
/-- The Cauchy formula for the derivative of a holomorphic function. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba
@@ -150,7 +150,7 @@ theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f
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 :=
+ ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀ :=
by
-- We apply the removable singularity theorem and the Cauchy formula to `dslope f w₀`
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (nonempty_of_mem hw₀).ne_empty)
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -158,7 +158,7 @@ theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U :
zero_smul, sub_zero]
· refine' circleIntegral.integral_congr (pos_of_mem_ball hw₀).le fun z hz => _
simp only [dslope_of_ne, Metric.sphere_disjoint_ball.ne_of_mem hz hw₀, slope, ← smul_assoc, sq,
- mul_inv, Ne.def, not_false_iff, vsub_eq_sub, Algebra.id.smul_eq_mul]
+ mul_inv, Ne, not_false_iff, vsub_eq_sub, Algebra.id.smul_eq_mul]
set_option linter.uppercaseLean3 false in
#align complex.two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
have
, replace
and suffices
(#10640)
No changes to tactic file, it's just boring fixes throughout the library.
This follows on from #6964.
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@@ -35,8 +35,8 @@ theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ
(hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c) : AnalyticAt ℂ f c := by
rcases (nhdsWithin_hasBasis nhds_basis_closedBall _).mem_iff.1 hd with ⟨R, hR0, hRs⟩
lift R to ℝ≥0 using hR0.le
- replace hc : ContinuousOn f (closedBall c R)
- · refine' fun z hz => ContinuousAt.continuousWithinAt _
+ replace hc : ContinuousOn f (closedBall c R) := by
+ refine' fun z hz => ContinuousAt.continuousWithinAt _
rcases eq_or_ne z c with (rfl | hne)
exacts [hc, (hRs ⟨hz, hne⟩).continuousAt]
exact (hasFPowerSeriesOnBall_of_differentiable_off_countable (countable_singleton c) hc
This is the supremum of
along with some minor fixes from failures on nightly-testing as Mathlib master
is merged into it.
Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.
I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0
branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.
In particular this includes adjustments for the Lean PRs
We can get rid of all the
local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)
macros across Mathlib (and in any projects that want to write natural number powers of reals).
Changes the default behaviour of simp
to (config := {decide := false})
. This makes simp
(and consequentially norm_num
) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp
or norm_num
to decide
or rfl
, or adding (config := {decide := true})
.
This changed the behaviour of simp
so that simp [f]
will only unfold "fully applied" occurrences of f
. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true })
. We may in future add a syntax for this, e.g. simp [!f]
; please provide feedback! In the meantime, we have made the following changes:
(config := { unfoldPartialApp := true })
in some places, to recover the old behaviour@[eqns]
to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp
and Function.flip
.This change in Lean may require further changes down the line (e.g. adding the !f
syntax, and/or upstreaming the special treatment for Function.comp
and Function.flip
, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!
Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>
@@ -23,8 +23,6 @@ open TopologicalSpace Metric Set Filter Asymptotics Function
open scoped Topology Filter NNReal Real
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
universe u
variable {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
@@ -23,7 +23,7 @@ open TopologicalSpace Metric Set Filter Asymptotics Function
open scoped Topology Filter NNReal Real
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
universe u
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-import Mathlib.Analysis.Calculus.FDerivAnalytic
+import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
@@ -2,16 +2,13 @@
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.complex.removable_singularity
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Analysis.Calculus.FDerivAnalytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
+#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
/-!
# Removable singularity theorem
The unported dependencies are
algebra.order.module
init.core
linear_algebra.free_module.finite.rank
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
linear_algebra.free_module.rank
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file