`BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth` {x : ℝ} (hx : 0 < x) : Real.cosh (x / 2) / (2 * Real.sinh (x / 2)) = ∑' n : ℕ, ((n : ℝ) + 1

Proved

BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterThermalTemperatureCore.

BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth {x : ℝ} (hx : 0 < x) : Real.cosh (x / 2) / (2 * Real.sinh (x / 2)) = ∑' n : ℕ, ((n : ℝ) + 1 / 2) * thermalProb (boseEinstein x) n

Formalization note: Lean 4 identifier BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth.

Preamble

-- Generated from ChapterThermalTemperatureCore.lean — theorem BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth
import Definitions.Def_ChapterCoherentOccupation
import Mathlib
import Definitions.Def_ChapterThermalTemperatureCore
import Definitions.Def_ChapterBoseEinstein
import Definitions.Def_ChapterCoherentTemperature
open BookProof.ChapterBoseEinstein
open BookProof.ChapterCoherentTemperature
open BookProof.ChapterThermalTemperatureCore


noncomputable section


open BookProof.ChapterCoherentTemperature BookProof.ChapterCoherentOccupation
open BookProof.ChapterBoseEinstein

variable {r : ℝ}

Formal statement

theorem BookProof.ChapterThermalTemperatureCore.thermal_temperature_eq_energy_expectation_coth {x : ℝ} (hx : 0 < x) :
    Real.cosh (x / 2) / (2 * Real.sinh (x / 2))
      = ∑' n : ℕ, ((n : ℝ) + 1 / 2) * thermalProb (boseEinstein x) n := by sorry

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