File: Mathlib/NumberTheory/LSeries/Deriv.lean

full_name: LSeries.logMul
code: /-- The (point-wise) product of `log : ℕ → ℂ` with `f`. -/
noncomputable abbrev LSeries.logMul (f : ℕ → ℂ) (n : ℕ) : ℂ 
goal: ℂ

full_name: LSeries.hasDerivAt_term
code: /-- The derivative of the terms of an L-series. -/
lemma LSeries.hasDerivAt_term (f : ℕ → ℂ) (n : ℕ) (s : ℂ) :
    HasDerivAt (fun z ↦ term f z n) (-(term (logMul f) s n)) s 
goal: HasDerivAt (fun z => LSeries.term f z n) (-LSeries.term (LSeries.logMul f) s n) s

full_name: _private.Mathlib.NumberTheory.LSeries.Deriv.0.LSeries.LSeriesSummable_logMul_and_hasDerivAt
code: private lemma LSeries.LSeriesSummable_logMul_and_hasDerivAt {f : ℕ → ℂ} {s : ℂ}
    (h : abscissaOfAbsConv f < s.re) :
    LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s 
goal: LSeriesSummable (LSeries.logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (LSeries.logMul f) s) s

full_name: LSeries_hasDerivAt
code: /-- If `re s` is greater than the abscissa of absolute convergence of `f`, then the L-series
of `f` is differentiable with derivative the negative of the L-series of the point-wise
product of `log` with `f`. -/
lemma LSeries_hasDerivAt {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    HasDerivAt (LSeries f) (- LSeries (logMul f) s) s 
goal: HasDerivAt (LSeries f) (-LSeries (LSeries.logMul f) s) s

full_name: LSeries_deriv
code: /-- If `re s` is greater than the abscissa of absolute convergence of `f`, then
the derivative of this L-series at `s` is the negative of the L-series of `log * f`. -/
lemma LSeries_deriv {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    deriv (LSeries f) s = - LSeries (logMul f) s 
goal: deriv (LSeries f) s = -LSeries (LSeries.logMul f) s

full_name: LSeries_deriv_eqOn
code: /-- The derivative of the L-series of `f` agrees with the negative of the L-series of
`log * f` on the right half-plane of absolute convergence. -/
lemma LSeries_deriv_eqOn {f : ℕ → ℂ} :
    {s | abscissaOfAbsConv f < s.re}.EqOn (deriv (LSeries f)) (- LSeries (logMul f)) 
goal: Set.EqOn (deriv (LSeries f)) (-LSeries (LSeries.logMul f)) {s | LSeries.abscissaOfAbsConv f < ↑s.re}

full_name: LSeriesSummable_logMul_of_lt_re
code: /-- If the L-series of `f` is summable at `s` and `re s < re s'`, then the L-series of the
point-wise product of `log` with `f` is summable at `s'`. -/
lemma LSeriesSummable_logMul_of_lt_re {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    LSeriesSummable (logMul f) s 
goal: LSeriesSummable (LSeries.logMul f) s

full_name: LSeries.abscissaOfAbsConv_logMul
code: /-- The abscissa of absolute convergence of the point-wise product of `log` and `f`
is the same as that of `f`. -/
@[simp]
lemma LSeries.abscissaOfAbsConv_logMul {f : ℕ → ℂ} :
    abscissaOfAbsConv (logMul f) = abscissaOfAbsConv f 
goal: LSeries.abscissaOfAbsConv (LSeries.logMul f) = LSeries.abscissaOfAbsConv f

full_name: LSeries.absicssaOfAbsConv_logPowMul
code: /-- The abscissa of absolute convergence of the point-wise product of a power of `log` and `f`
is the same as that of `f`. -/
@[simp]
lemma LSeries.absicssaOfAbsConv_logPowMul {f : ℕ → ℂ} {m : ℕ} :
    abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f 
goal: LSeries.abscissaOfAbsConv (LSeries.logMul^[m] f) = LSeries.abscissaOfAbsConv f

full_name: LSeries_iteratedDeriv
code: /-- If `re s` is greater than the abscissa of absolute convergence of `f`, then
the `m`th derivative of this L-series is `(-1)^m` times the L-series of `log^m * f`. -/
lemma LSeries_iteratedDeriv {f : ℕ → ℂ} (m : ℕ) {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    iteratedDeriv m (LSeries f) s = (-1) ^ m * LSeries (logMul^[m] f) s 
goal: iteratedDeriv m (LSeries f) s = (-1) ^ m * LSeries (LSeries.logMul^[m] f) s

full_name: LSeries_differentiableOn
code: /-- The L-series of `f` is complex differentiable in its open half-plane of absolute
convergence. -/
lemma LSeries_differentiableOn (f : ℕ → ℂ) :
    DifferentiableOn ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} 
goal: DifferentiableOn ℂ (LSeries f) {s | LSeries.abscissaOfAbsConv f < ↑s.re}

full_name: LSeries_analyticOn
code: /-- The L-series of `f` is holomorphic on its open half-plane of absolute convergence. -/
lemma LSeries_analyticOn (f : ℕ → ℂ) :
    AnalyticOn ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} 
goal: AnalyticOn ℂ (LSeries f) {s | LSeries.abscissaOfAbsConv f < ↑s.re}