English
Let p ≥ 1, q > −2 and b > 0. Then ∫ z∈ℂ ‖z‖^q e^(−b‖z‖^p) d z = (2π/p) b^{−(q+2)/p} Γ((q+2)/p).
Русский
Пусть p ≥ 1, q > −2 и b > 0. Тогда интеграл по комплексной плоскости от |z|^q e^{−b|z|^p} dz равен (2π/p) b^{−(q+2)/p} Γ((q+2)/p).
LaTeX
$$$ \displaystyle \int_{\mathbb{C}} \|z\|^{q} e^{-b\|z\|^{p}} \, dz = \frac{2\pi}{p} b^{-(q+2)/p} \Gamma\left(\frac{q+2}{p}\right), \quad p\ge 1, q> -2, b>0 $$$
Lean4
/-- If `μ` is a finite measure and the indicators of measurable sets `Aᵢ` tend pointwise to
the indicator of a set `A`, then the measures `μ Aᵢ` tend to the measure `μ A`. -/
theorem tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure (μ : Measure α) [IsFiniteMeasure μ]
(As_mble : ∀ i, MeasurableSet (As i)) (h_lim : ∀ x, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) :=
by
rcases L.eq_or_neBot with rfl | _
· exact tendsto_bot
apply tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure L ?_ As_mble (ae_of_all μ h_lim)
exact measurableSet_of_tendsto_indicator L As_mble h_lim
