File: Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean

full_name: WeakFEPair.f
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: ℝ → E

full_name: WeakFEPair
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: Type u_1

full_name: WeakFEPair.hg_int
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: MeasureTheory.LocallyIntegrableOn self.g (Set.Ioi 0) MeasureTheory.volume

full_name: WeakFEPair.h_feq
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: x ∈ Set.Ioi 0 → self.f (1 / x) = (self.ε * ↑(x ^ self.k)) • self.g x

full_name: WeakFEPair.ε
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: ℂ

full_name: WeakFEPair.g₀
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: E

full_name: WeakFEPair.hf_top
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: (fun x => self.f x - self.f₀) =O[Filter.atTop] fun x => x ^ r

full_name: WeakFEPair.f₀
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: E

full_name: WeakFEPair.hg_top
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: (fun x => self.g x - self.g₀) =O[Filter.atTop] fun x => x ^ r

full_name: WeakFEPair.hε
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: self.ε ≠ 0

full_name: WeakFEPair.g
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: ℝ → E

full_name: WeakFEPair.hk
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: 0 < self.k

full_name: WeakFEPair.k
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: ℝ

full_name: WeakFEPair.hf_int
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(most general version: rapid decay at `∞` up to constant terms) -/
structure WeakFEPair where
  /-- The functions whose Mellin transform we study -/
  (f g : ℝ → E)
  /-- Weight (exponent in the functional equation) -/
  (k : ℝ)
  /-- Root number -/
  (ε : ℂ)
  /-- Constant terms at `∞` -/
  (f₀ g₀ : E)
  (hf_int : LocallyIntegrableOn f (Ioi 0))
  (hg_int : LocallyIntegrableOn g (Ioi 0))
  (hk : 0 < k)
  (hε : ε ≠ 0)
  (h_feq : ∀ x ∈ Ioi 0, f (1 / x) = (ε * ↑(x ^ k)) • g x)
  (hf_top (r : ℝ) : (f · - f₀) =O[atTop] (· ^ r))
  (hg_top (r : ℝ) : (g · - g₀) =O[atTop] (· ^ r))
goal: MeasureTheory.LocallyIntegrableOn self.f (Set.Ioi 0) MeasureTheory.volume

full_name: StrongFEPair.hg₀
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(version without constant terms) -/
structure StrongFEPair extends WeakFEPair E where (hf₀ : f₀ = 0) (hg₀ : g₀ = 0)
goal: self.g₀ = 0

full_name: StrongFEPair
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(version without constant terms) -/
structure StrongFEPair extends WeakFEPair E where (hf₀ : f₀ = 0) (hg₀ : g₀ = 0)
goal: Type u_1

full_name: StrongFEPair.hf₀
code: /-- A structure designed to hold the hypotheses for the Mellin-functional-equation argument
(version without constant terms) -/
structure StrongFEPair extends WeakFEPair E where (hf₀ : f₀ = 0) (hg₀ : g₀ = 0)
goal: self.f₀ = 0

full_name: WeakFEPair.h_feq'
code: /-- Reformulated functional equation with `f` and `g` interchanged. -/
lemma WeakFEPair.h_feq' (P : WeakFEPair E) (x : ℝ) (hx : 0 < x) :
    P.g (1 / x) = (P.ε⁻¹ * ↑(x ^ P.k)) • P.f x 
goal: P.g (1 / x) = (P.ε⁻¹ * ↑(x ^ P.k)) • P.f x

full_name: WeakFEPair.symm
code: /-- The hypotheses are symmetric in `f` and `g`, with the constant `ε` replaced by `ε⁻¹`. -/
def WeakFEPair.symm (P : WeakFEPair E) : WeakFEPair E 
goal: WeakFEPair E

full_name: StrongFEPair.symm
code: /-- The hypotheses are symmetric in `f` and `g`, with the constant `ε` replaced by `ε⁻¹`. -/
def StrongFEPair.symm (P : StrongFEPair E) : StrongFEPair E 
goal: StrongFEPair E

full_name: WeakFEPair.hf_zero
code: /-- As `x → 0`, we have `f x = x ^ (-P.k) • constant` up to a rapidly decaying error. -/
lemma hf_zero (P : WeakFEPair E) (r : ℝ) :
    (fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] (· ^ r) 
goal: (fun x => P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] fun x => x ^ r

full_name: WeakFEPair.hf_zero'
code: /-- Power asymptotic for `f - f₀` as `x → 0`. -/
lemma hf_zero' (P : WeakFEPair E) :
    (fun x : ℝ ↦ P.f x - P.f₀) =O[𝓝[>] 0] (· ^ (-P.k)) 
goal: (fun x => P.f x - P.f₀) =O[𝓝[>] 0] fun x => x ^ (-P.k)

full_name: StrongFEPair.hf_top'
code: /-- As `x → ∞`, `f x` decays faster than any power of `x`. -/
lemma hf_top' (r : ℝ) : P.f =O[atTop] (· ^ r) 
goal: P.f =O[Filter.atTop] fun x => x ^ r

full_name: StrongFEPair.hf_zero'
code: /-- As `x → 0`, `f x` decays faster than any power of `x`. -/
lemma hf_zero' (r : ℝ) : P.f =O[𝓝[>] 0] (· ^ r) 
goal: P.f =O[𝓝[>] 0] fun x => x ^ r

full_name: StrongFEPair.Λ
code: /-- The completed L-function. -/
def Λ : ℂ → E 
goal: ℂ → E

full_name: StrongFEPair.hasMellin
code: /-- The Mellin transform of `f` is well-defined and equal to `P.Λ s`, for all `s`. -/
theorem hasMellin (s : ℂ) : HasMellin P.f s (P.Λ s) 
goal: HasMellin P.f s (P.Λ s)

full_name: StrongFEPair.Λ_eq
code: lemma Λ_eq : P.Λ = mellin P.f 
goal: P.Λ = mellin P.f

full_name: StrongFEPair.symm_Λ_eq
code: lemma symm_Λ_eq : P.symm.Λ = mellin P.g 
goal: P.symm.Λ = mellin P.g

full_name: StrongFEPair.differentiable_Λ
code: /-- If `(f, g)` are a strong FE pair, then the Mellin transform of `f` is entire. -/
theorem differentiable_Λ : Differentiable ℂ P.Λ 
goal: Differentiable ℂ P.Λ

full_name: StrongFEPair.functional_equation
code: /-- Main theorem about strong FE pairs: if `(f, g)` are a strong FE pair, then the Mellin
transforms of `f` and `g` are related by `s ↦ k - s`.

This is proved by making a substitution `t ↦ t⁻¹` in the Mellin transform integral. -/
theorem functional_equation (s : ℂ) :
    P.Λ (P.k - s) = P.ε • P.symm.Λ s 
goal: P.Λ (↑P.k - s) = P.ε • P.symm.Λ s

full_name: WeakFEPair.f_modif
code: /-- Piecewise modified version of `f` with optimal asymptotics. We deliberately choose intervals
which don't quite join up, so the function is `0` at `x = 1`, in order to maintain symmetry;
there is no "good" choice of value at `1`. -/
def f_modif : ℝ → E 
goal: ℝ → E

full_name: WeakFEPair.g_modif
code: /-- Piecewise modified version of `g` with optimal asymptotics. -/
def g_modif : ℝ → E 
goal: ℝ → E

full_name: WeakFEPair.hf_modif_int
code: lemma hf_modif_int :
    LocallyIntegrableOn P.f_modif (Ioi 0) 
goal: MeasureTheory.LocallyIntegrableOn P.f_modif (Set.Ioi 0) MeasureTheory.volume

full_name: WeakFEPair.hf_modif_FE
code: lemma hf_modif_FE (x : ℝ) (hx : 0 < x) :
    P.f_modif (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x 
goal: P.f_modif (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x

full_name: WeakFEPair.toStrongFEPair
code: /-- Given a weak FE-pair `(f, g)`, modify it into a strong FE-pair by subtracting suitable
correction terms from `f` and `g`. -/
def toStrongFEPair : StrongFEPair E 
goal: StrongFEPair E

full_name: WeakFEPair.f_modif_aux1
code: lemma f_modif_aux1 : EqOn (fun x ↦ P.f_modif x - P.f x + P.f₀)
    ((Ioo 0 1).indicator (fun x : ℝ ↦ P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀)
    + ({1} : Set ℝ).indicator (fun _ ↦ P.f₀ - P.f 1)) (Ioi 0) 
goal: Set.EqOn (fun x => P.f_modif x - P.f x + P.f₀)
  (((Set.Ioo 0 1).indicator fun x => P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀) + {1}.indicator fun x => P.f₀ - P.f 1)
  (Set.Ioi 0)

full_name: WeakFEPair.f_modif_aux2
code: /-- Compute the Mellin transform of the modifying term used to kill off the constants at
`0` and `∞`. -/
lemma f_modif_aux2 {s : ℂ} (hs : P.k < re s) :
    mellin (fun x ↦ P.f_modif x - P.f x + P.f₀) s = (1 / s) • P.f₀ + (P.ε  / (P.k - s)) • P.g₀ 
goal: mellin (fun x => P.f_modif x - P.f x + P.f₀) s = (1 / s) • P.f₀ + (P.ε / (↑P.k - s)) • P.g₀

full_name: WeakFEPair.Λ₀
code: /-- An entire function which differs from the Mellin transform of `f - f₀`, where defined, by a
correction term of the form `A / s + B / (k - s)`. -/
def Λ₀ : ℂ → E 
goal: ℂ → E

full_name: WeakFEPair.Λ
code: /-- A meromorphic function which agrees with the Mellin transform of `f - f₀` where defined -/
def Λ (s : ℂ) : E 
goal: E

full_name: WeakFEPair.Λ₀_eq
code: lemma Λ₀_eq (s : ℂ) : P.Λ₀ s = P.Λ s + (1 / s) • P.f₀ + (P.ε / (P.k - s)) • P.g₀ 
goal: P.Λ₀ s = P.Λ s + (1 / s) • P.f₀ + (P.ε / (↑P.k - s)) • P.g₀

full_name: WeakFEPair.symm_Λ₀_eq
code: lemma symm_Λ₀_eq (s : ℂ) :
    P.symm.Λ₀ s = P.symm.Λ s + (1 / s) • P.g₀ + (P.ε⁻¹ / (P.k - s)) • P.f₀ 
goal: P.symm.Λ₀ s = P.symm.Λ s + (1 / s) • P.g₀ + (P.ε⁻¹ / (↑P.k - s)) • P.f₀

full_name: WeakFEPair.differentiable_Λ₀
code: theorem differentiable_Λ₀ : Differentiable ℂ P.Λ₀ 
goal: Differentiable ℂ P.Λ₀

full_name: WeakFEPair.differentiableAt_Λ
code: theorem differentiableAt_Λ {s : ℂ} (hs : s ≠ 0 ∨ P.f₀ = 0) (hs' : s ≠ P.k ∨ P.g₀ = 0) :
    DifferentiableAt ℂ P.Λ s 
goal: DifferentiableAt ℂ P.Λ s

full_name: WeakFEPair.hasMellin
code: /-- Relation between `Λ s` and the Mellin transform of `f - f₀`, where the latter is defined. -/
theorem hasMellin {s : ℂ} (hs : P.k < s.re) : HasMellin (P.f · - P.f₀) s (P.Λ s) 
goal: HasMellin (fun x => P.f x - P.f₀) s (P.Λ s)

full_name: WeakFEPair.functional_equation₀
code: /-- Functional equation formulated for `Λ₀`. -/
theorem functional_equation₀ (s : ℂ) : P.Λ₀ (P.k - s) = P.ε • P.symm.Λ₀ s 
goal: P.Λ₀ (↑P.k - s) = P.ε • P.symm.Λ₀ s

full_name: WeakFEPair.functional_equation
code: /-- Functional equation formulated for `Λ`. -/
theorem functional_equation (s : ℂ) :
    P.Λ (P.k - s) = P.ε • P.symm.Λ s 
goal: P.Λ (↑P.k - s) = P.ε • P.symm.Λ s

full_name: WeakFEPair.Λ_residue_k
code: /-- The residue of `Λ` at `s = k` is equal to `ε • g₀`. -/
theorem Λ_residue_k :
    Tendsto (fun s : ℂ ↦ (s - P.k) • P.Λ s) (𝓝[≠] P.k) (𝓝 (P.ε • P.g₀)) 
goal: Filter.Tendsto (fun s => (s - ↑P.k) • P.Λ s) (𝓝[≠] ↑P.k) (𝓝 (P.ε • P.g₀))

full_name: WeakFEPair.Λ_residue_zero
code: /-- The residue of `Λ` at `s = 0` is equal to `-f₀`. -/
theorem Λ_residue_zero :
    Tendsto (fun s : ℂ ↦ s • P.Λ s) (𝓝[≠] 0) (𝓝 (-P.f₀)) 
goal: Filter.Tendsto (fun s => s • P.Λ s) (𝓝[≠] 0) (𝓝 (-P.f₀))