English
For E a normed space, the integral over the product polar target of the first-component weights equals the integral of f, for f: (ι → ℂ) → E.
Русский
Для евклидового пространства E справедлива формула интеграла по полярной симм-координате: весовая сумма по первым координатам равна интегралу f.
LaTeX
$$$\int_{p \in (\mathrm{Set.univ.pi}\; \mathrm{polarCoord.target})} \Big(\prod_i (p_i)_1\Big) \cdot f(\mathrm{polarCoord.symm}(p_i)) \, dp = \int_p f(p) \, dp.$$$
Lean4
theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
induction p using Polynomial.induction_on' with
| monomial n c =>
simpa [exp_neg, div_eq_mul_inv, mul_assoc] using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)
| add p q hp hq => simpa [add_div] using hp.add hq
