Integrals with exponential decay at ∞ #

As easy special cases of general theorems in the library, we prove the following test for integrability:

integrable_of_is_O_exp_neg: If f is continuous on [a,∞), for some a ∈ ℝ, and there exists b > 0 such that f(x) = O(exp(-b x)) as x → ∞, then f is integrable on (a, ∞).

theorem integral_exp_neg_le {b : ℝ} (a X : ℝ) (h2 : 0 < b) :

∫ (x : ℝ) in a..X, real.exp (-b * x) ≤ real.exp (-b * a) / b

Integral of exp (-b * x) over (a, X) is bounded as X → ∞.

theorem exp_neg_integrable_on_Ioi (a : ℝ) {b : ℝ} (h : 0 < b) :

exp (-b * x) is integrable on (a, ∞).

theorem integrable_of_is_O_exp_neg {f : ℝ → ℝ} {a b : ℝ} (h0 : 0 < b) (h1 : continuous_on f (set.Ici a)) (h2 : f =O[filter.at_top ] λ (x : ℝ), real.exp (-b * x)) :

If f is continuous on [a, ∞), and is O (exp (-b * x)) at ∞ for some b > 0, then f is integrable on (a, ∞).