Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Sum eq integral add integral deriv

sum_eq_integral_add_integral_deriv

analytic number theory prime numbers asymptotics euler maclaurin

Plain-language statement

A first-order Euler–Maclaurin formula. For $0\le a\le b$ and a differentiable function $f$ whose derivative is continuous on $[a,b]$ , the sum over integers $\lfloor a\rfloor<k\le\lfloor b\rfloor$ equals $f(a)B_1(a)-f(b)B_1(b)+\int_a^b f(t)\,dt+\int_a^b f'(t)B_1(t)\,dt,$ where $B_1(t)=t-\lfloor t\rfloor-\tfrac12$ .

Exact Lean statement

theorem sum_eq_integral_add_integral_deriv (ha : 0 ≤ a) (hab : a ≤ b)
    (hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t)
    (h_cont : ContinuousOn (deriv f) [[a, b]]) :
    ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f k =
      f a * B1 a - f b * B1 b + (∫ t in a..b, f t) + ∫ t in a..b, deriv f t * B1 t

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem sum_eq_integral_add_integral_deriv (ha : 0 ≤ a) (hab : a ≤ b)    (hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t)    (h_cont : ContinuousOn (deriv f) [[a, b]]) :    ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f k =      f a * B1 a - f b * B1 b + (∫ t in a..b, f t) + ∫ t in a..b, deriv f t * B1 t  := by  have := sum_mul_eq_sub_sub_integral_mul (fun _ ↦ 1) ha hab hf_diff (Set.uIcc_of_le hab ▸ h_cont).integrableOn_Icc  simp only [mul_one, sum_const, Nat.card_Icc, tsub_zero, nsmul_eq_mul, Nat.cast_add,    Nat.cast_one] at this  rw [this, ← intervalIntegral.integral_of_le hab]  rw [integral_deriv_mul_floor_add_one ha hab hf_diff h_cont]  unfold B1  push_cast  ring

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/EulerMaclaurin.lean:67-79

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Admissible bound mono

admissible_bound.mono

Plain-language statement

For positive parameters $A,B,C,R$ , the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero

AnalyticOn_divRemovable_zero

Plain-language statement

Let $f$ be analytic on an open set $s$ containing $0$ , and suppose $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero closed Ball

AnalyticOn_divRemovable_zero_closedBall

Plain-language statement

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then $g$ is analytic on the entire closed disc, including at the removed singularity.

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record