English
A formal expansion shows that the product of exponentials corresponds to the exponential of the sum under the Fourier transform, matching the standard identity exp(x)exp(y) = exp(x+y).
Русский
Формальное разложение показывает, что произведение экспонент переходит к экспоненте суммы под преобразованием Фурье.
LaTeX
$$$$\\mathcal{F}\\bigl( e^{x} e^{y} \\bigr) = \\mathcal{F}\\bigl( e^{x+y} \\bigr).$$$$
Lean4
theorem exp_neg_mul_rpow_isLittleO_exp_neg {p b : ℝ} (hb : 0 < b) (hp : 1 < p) :
(fun x : ℝ => exp (-b * x ^ p)) =o[atTop] fun x : ℝ => exp (-x) :=
by
rw [isLittleO_exp_comp_exp_comp]
suffices Tendsto (fun x => x * (b * x ^ (p - 1) + -1)) atTop atTop
by
refine Tendsto.congr' ?_ this
refine eventuallyEq_of_mem (Ioi_mem_atTop (0 : ℝ)) (fun x hx => ?_)
rw [mem_Ioi] at hx
rw [rpow_sub_one hx.ne']
field_simp
ring
apply tendsto_id.atTop_mul_atTop₀
refine tendsto_atTop_add_const_right atTop (-1 : ℝ) ?_
exact Tendsto.const_mul_atTop hb (tendsto_rpow_atTop (by linarith))
