mathlib documentation

analysis.asymptotics.specific_asymptotics

A collection of specific asymptotic results #

This file contains specific lemmas about asymptotics which don't have their place in the general theory developped in analysis.asymptotics.asymptotics.

theorem filter.is_bounded_under.is_o_sub_self_inv {๐•œ : Type u_1} {E : Type u_2} [normed_field ๐•œ] [has_norm E] {a : ๐•œ} {f : ๐•œ โ†’ E} (h : filter.is_bounded_under has_le.le (nhds_within a {a}แถœ) (has_norm.norm โˆ˜ f)) :
f =o[nhds_within a {a}แถœ] ฮป (x : ๐•œ), (x - a)โปยน

If f : ๐•œ โ†’ E is bounded in a punctured neighborhood of a, then f(x) = o((x - a)โปยน) as x โ†’ a, x โ‰  a.

theorem pow_div_pow_eventually_eq_at_top {๐•œ : Type u_1} [linear_ordered_field ๐•œ] {p q : โ„•} :
(ฮป (x : ๐•œ), x ^ p / x ^ q) =แถ [filter.at_top] ฮป (x : ๐•œ), x ^ (โ†‘p - โ†‘q)
theorem pow_div_pow_eventually_eq_at_bot {๐•œ : Type u_1} [linear_ordered_field ๐•œ] {p q : โ„•} :
(ฮป (x : ๐•œ), x ^ p / x ^ q) =แถ [filter.at_bot] ฮป (x : ๐•œ), x ^ (โ†‘p - โ†‘q)
theorem tendsto_zpow_at_top_at_top {๐•œ : Type u_1} [linear_ordered_field ๐•œ] {n : โ„ค} (hn : 0 < n) :
filter.tendsto (ฮป (x : ๐•œ), x ^ n) filter.at_top filter.at_top
theorem tendsto_pow_div_pow_at_top_at_top {๐•œ : Type u_1} [linear_ordered_field ๐•œ] {p q : โ„•} (hpq : q < p) :
filter.tendsto (ฮป (x : ๐•œ), x ^ p / x ^ q) filter.at_top filter.at_top
theorem tendsto_pow_div_pow_at_top_zero {๐•œ : Type u_1} [linear_ordered_field ๐•œ] [topological_space ๐•œ] [order_topology ๐•œ] {p q : โ„•} (hpq : p < q) :
filter.tendsto (ฮป (x : ๐•œ), x ^ p / x ^ q) filter.at_top (nhds 0)
theorem asymptotics.is_o_pow_pow_at_top_of_lt {๐•œ : Type u_1} [normed_linear_ordered_field ๐•œ] [order_topology ๐•œ] {p q : โ„•} (hpq : p < q) :
(ฮป (x : ๐•œ), x ^ p) =o[filter.at_top] ฮป (x : ๐•œ), x ^ q
theorem asymptotics.is_O.trans_tendsto_norm_at_top {๐•œ : Type u_1} [normed_linear_ordered_field ๐•œ] {ฮฑ : Type u_2} {u v : ฮฑ โ†’ ๐•œ} {l : filter ฮฑ} (huv : u =O[l] v) (hu : filter.tendsto (ฮป (x : ฮฑ), โˆฅu xโˆฅ) l filter.at_top) :
filter.tendsto (ฮป (x : ฮฑ), โˆฅv xโˆฅ) l filter.at_top
theorem asymptotics.is_o.sum_range {ฮฑ : Type u_1} [normed_group ฮฑ] {f : โ„• โ†’ ฮฑ} {g : โ„• โ†’ โ„} (h : f =o[filter.at_top] g) (hg : 0 โ‰ค g) (h'g : filter.tendsto (ฮป (n : โ„•), (finset.range n).sum (ฮป (i : โ„•), g i)) filter.at_top filter.at_top) :
(ฮป (n : โ„•), (finset.range n).sum (ฮป (i : โ„•), f i)) =o[filter.at_top] ฮป (n : โ„•), (finset.range n).sum (ฮป (i : โ„•), g i)
theorem asymptotics.is_o_sum_range_of_tendsto_zero {ฮฑ : Type u_1} [normed_group ฮฑ] {f : โ„• โ†’ ฮฑ} (h : filter.tendsto f filter.at_top (nhds 0)) :
(ฮป (n : โ„•), (finset.range n).sum (ฮป (i : โ„•), f i)) =o[filter.at_top] ฮป (n : โ„•), โ†‘n
theorem filter.tendsto.cesaro_smul {E : Type u_1} [normed_group E] [normed_space โ„ E] {u : โ„• โ†’ E} {l : E} (h : filter.tendsto u filter.at_top (nhds l)) :

The Cesaro average of a converging sequence converges to the same limit.

theorem filter.tendsto.cesaro {u : โ„• โ†’ โ„} {l : โ„} (h : filter.tendsto u filter.at_top (nhds l)) :
filter.tendsto (ฮป (n : โ„•), (โ†‘n)โปยน * (finset.range n).sum (ฮป (i : โ„•), u i)) filter.at_top (nhds l)

The Cesaro average of a converging sequence converges to the same limit.