integrableOn_Ioi_comp_mul_right_iff

English

Let f′ and g be two vector-valued functions and L a bilinear form; under suitable integrability and growth conditions, the integral of L(u x, v' x) + L(u' x, v x) over the real line equals n − m, where m and n bound the bilinear term at infinities.

Русский

Пусть f′ и g — векторнозначные функции, L — билинеарная форма; при подходящих условиях интегрируемости и роста интеграл L(u x, v' x) + L(u' x, v x) по ℝ равен n − m.

LaTeX

$$$$ \int_{-\infty}^{\infty} \bigl( L(u x, v' x) + L(u' x, v x) \bigr) \, dx = n - m, $$$$

Lean4

theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
    IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) := by
  simpa only [mul_comm, mul_zero] using integrableOn_Ioi_comp_mul_left_iff f c ha

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