File: Mathlib/Analysis/Analytic/Meromorphic.lean

full_name: MeromorphicAt
code: /-- Meromorphy of `f` at `x` (more precisely, on a punctured neighbourhood of `x`; the value at
`x` itself is irrelevant). -/
def MeromorphicAt (f : 𝕜 → E) (x : 𝕜) 
goal: Prop

full_name: AnalyticAt.meromorphicAt
code: lemma AnalyticAt.meromorphicAt {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x) :
    MeromorphicAt f x 
goal: MeromorphicAt f x

full_name: MeromorphicAt.id
code: lemma id (x : 𝕜) : MeromorphicAt id x 
goal: MeromorphicAt id x

full_name: MeromorphicAt.const
code: lemma const (e : E) (x : 𝕜) : MeromorphicAt (fun _ ↦ e) x 
goal: MeromorphicAt (fun x => e) x

full_name: MeromorphicAt.add
code: lemma add {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f + g) x 
goal: MeromorphicAt (f + g) x

full_name: MeromorphicAt.smul
code: lemma smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f • g) x 
goal: MeromorphicAt (f • g) x

full_name: MeromorphicAt.mul
code: lemma mul {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f * g) x 
goal: MeromorphicAt (f * g) x

full_name: MeromorphicAt.neg
code: lemma neg {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicAt (-f) x 
goal: MeromorphicAt (-f) x

full_name: MeromorphicAt.neg_iff
code: @[simp]
lemma neg_iff {f : 𝕜 → E} {x : 𝕜} :
    MeromorphicAt (-f) x ↔ MeromorphicAt f x 
goal: MeromorphicAt (-f) x ↔ MeromorphicAt f x

full_name: MeromorphicAt.sub
code: lemma sub {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f - g) x 
goal: MeromorphicAt (f - g) x

full_name: MeromorphicAt.congr
code: /-- With our definitions, `MeromorphicAt f x` depends only on the values of `f` on a punctured
neighbourhood of `x` (not on `f x`) -/
lemma congr {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hfg : f =ᶠ[𝓝[≠] x] g) :
    MeromorphicAt g x 
goal: MeromorphicAt g x

full_name: MeromorphicAt.inv
code: lemma inv {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicAt f⁻¹ x 
goal: MeromorphicAt f⁻¹ x

full_name: MeromorphicAt.inv_iff
code: @[simp]
lemma inv_iff {f : 𝕜 → 𝕜} {x : 𝕜} :
    MeromorphicAt f⁻¹ x ↔ MeromorphicAt f x 
goal: MeromorphicAt f⁻¹ x ↔ MeromorphicAt f x

full_name: MeromorphicAt.div
code: lemma div {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f / g) x 
goal: MeromorphicAt (f / g) x

full_name: MeromorphicAt.pow
code: lemma pow {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℕ) : MeromorphicAt (f ^ n) x 
goal: MeromorphicAt (f ^ n) x

full_name: MeromorphicAt.zpow
code: lemma zpow {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : MeromorphicAt (f ^ n) x 
goal: MeromorphicAt (f ^ n) x

full_name: MeromorphicAt.order
code: /-- The order of vanishing of a meromorphic function, as an element of `ℤ ∪ ∞` (to include the
case of functions identically 0 near `x`). -/
noncomputable def order {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) : WithTop ℤ 
goal: WithTop ℤ

full_name: MeromorphicAt.order_eq_top_iff
code: lemma order_eq_top_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) :
    hf.order = ⊤ ↔ ∀ᶠ z in 𝓝[≠] x, f z = 0 
goal: hf.order = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

full_name: MeromorphicAt.order_eq_int_iff
code: lemma order_eq_int_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : hf.order = n ↔
    ∃ g : 𝕜 → E, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z 
goal: hf.order = ↑n ↔ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z

full_name: AnalyticAt.meromorphicAt_order
code: /-- Compatibility of notions of `order` for analytic and meromorphic functions. -/
lemma _root_.AnalyticAt.meromorphicAt_order {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x) :
    hf.meromorphicAt.order = hf.order.map (↑) 
goal: ⋯.order = WithTop.map Nat.cast hf.order

full_name: MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt
code: lemma iff_eventuallyEq_zpow_smul_analyticAt {f : 𝕜 → E} {x : 𝕜} : MeromorphicAt f x ↔
    ∃ (n : ℤ) (g : 𝕜 → E), AnalyticAt 𝕜 g x ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z 
goal: MeromorphicAt f x ↔ ∃ n g, AnalyticAt 𝕜 g x ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z

full_name: MeromorphicOn
code: /-- Meromorphy of a function on a set. -/
def MeromorphicOn (f : 𝕜 → E) (U : Set 𝕜) : Prop 
goal: Prop

full_name: AnalyticOn.meromorphicOn
code: lemma AnalyticOn.meromorphicOn {f : 𝕜 → E} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) :
    MeromorphicOn f U 
goal: MeromorphicOn f U

full_name: MeromorphicOn.id
code: lemma id {U : Set 𝕜} : MeromorphicOn id U 
goal: MeromorphicOn id U

full_name: MeromorphicOn.const
code: lemma const (e : E) {U : Set 𝕜} : MeromorphicOn (fun _ ↦ e) U 
goal: MeromorphicOn (fun x => e) U

full_name: MeromorphicOn.mono_set
code: lemma mono_set {V : Set 𝕜} (hv : V ⊆ U) : MeromorphicOn f V 
goal: MeromorphicOn f V

full_name: MeromorphicOn.add
code: lemma add : MeromorphicOn (f + g) U 
goal: MeromorphicOn (f + g) U

full_name: MeromorphicOn.sub
code: lemma sub : MeromorphicOn (f - g) U 
goal: MeromorphicOn (f - g) U

full_name: MeromorphicOn.neg
code: lemma neg : MeromorphicOn (-f) U 
goal: MeromorphicOn (-f) U

full_name: MeromorphicOn.neg_iff
code: @[simp] lemma neg_iff : MeromorphicOn (-f) U ↔ MeromorphicOn f U 
goal: MeromorphicOn (-f) U ↔ MeromorphicOn f U

full_name: MeromorphicOn.smul
code: lemma smul : MeromorphicOn (s • f) U 
goal: MeromorphicOn (s • f) U

full_name: MeromorphicOn.mul
code: lemma mul : MeromorphicOn (s * t) U 
goal: MeromorphicOn (s * t) U

full_name: MeromorphicOn.inv
code: lemma inv : MeromorphicOn s⁻¹ U 
goal: MeromorphicOn s⁻¹ U

full_name: MeromorphicOn.inv_iff
code: @[simp] lemma inv_iff : MeromorphicOn s⁻¹ U ↔ MeromorphicOn s U 
goal: MeromorphicOn s⁻¹ U ↔ MeromorphicOn s U

full_name: MeromorphicOn.div
code: lemma div : MeromorphicOn (s / t) U 
goal: MeromorphicOn (s / t) U

full_name: MeromorphicOn.pow
code: lemma pow (n : ℕ) : MeromorphicOn (s ^ n) U 
goal: MeromorphicOn (s ^ n) U

full_name: MeromorphicOn.zpow
code: lemma zpow (n : ℤ) : MeromorphicOn (s ^ n) U 
goal: MeromorphicOn (s ^ n) U

full_name: MeromorphicOn.congr
code: lemma congr (h_eq : Set.EqOn f g U) (hu : IsOpen U) : MeromorphicOn g U 
goal: MeromorphicOn g U