statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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tendsto_rpow_neg_at_top {y : ℝ} (hy : 0 < y) : tendsto (λ x : ℝ, x ^ (-y)) at_top (𝓝 0) | tendsto.congr' (eventually_eq_of_mem (Ioi_mem_at_top 0) (λ x hx, (rpow_neg (le_of_lt hx) y).symm))
(tendsto_rpow_at_top hy).inv_tendsto_at_top | lemma | tendsto_rpow_neg_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"tendsto_rpow_at_top"
] | The function `x ^ (-y)` tends to `0` at `+∞` for any positive real `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_rpow_div_mul_add (a b c : ℝ) (hb : 0 ≠ b) :
tendsto (λ x, x ^ (a / (b*x+c))) at_top (𝓝 1) | begin
refine tendsto.congr' _ ((tendsto_exp_nhds_0_nhds_1.comp
(by simpa only [mul_zero, pow_one] using ((@tendsto_const_nhds _ _ _ a _).mul
(tendsto_div_pow_mul_exp_add_at_top b c 1 hb)))).comp tendsto_log_at_top),
apply eventually_eq_of_mem (Ioi_mem_at_top (0:ℝ)),
intros x hx,
simp only [set.mem_Ioi... | lemma | tendsto_rpow_div_mul_add | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"mul_zero",
"pow_one",
"set.mem_Ioi",
"tendsto_const_nhds"
] | The function `x ^ (a / (b * x + c))` tends to `1` at `+∞`, for any real numbers `a`, `b`, and
`c` such that `b` is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_rpow_div : tendsto (λ x, x ^ ((1:ℝ) / x)) at_top (𝓝 1) | by { convert tendsto_rpow_div_mul_add (1:ℝ) _ (0:ℝ) zero_ne_one, funext, congr' 2, ring } | lemma | tendsto_rpow_div | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"ring",
"tendsto_rpow_div_mul_add",
"zero_ne_one"
] | The function `x ^ (1 / x)` tends to `1` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_rpow_neg_div : tendsto (λ x, x ^ (-(1:ℝ) / x)) at_top (𝓝 1) | by { convert tendsto_rpow_div_mul_add (-(1:ℝ)) _ (0:ℝ) zero_ne_one, funext, congr' 2, ring } | lemma | tendsto_rpow_neg_div | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"ring",
"tendsto_rpow_div_mul_add",
"zero_ne_one"
] | The function `x ^ (-1 / x)` tends to `1` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_div_rpow_at_top (s : ℝ) : tendsto (λ x : ℝ, exp x / x ^ s) at_top at_top | begin
cases archimedean_iff_nat_lt.1 (real.archimedean) s with n hn,
refine tendsto_at_top_mono' _ _ (tendsto_exp_div_pow_at_top n),
filter_upwards [eventually_gt_at_top (0 : ℝ), eventually_ge_at_top (1 : ℝ)] with x hx₀ hx₁,
rw [div_le_div_left (exp_pos _) (pow_pos hx₀ _) (rpow_pos_of_pos hx₀ _), ←rpow_nat_cast... | lemma | tendsto_exp_div_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_le_div_left",
"exp",
"pow_pos"
] | The function `exp(x) / x ^ s` tends to `+∞` at `+∞`, for any real number `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_mul_div_rpow_at_top (s : ℝ) (b : ℝ) (hb : 0 < b) :
tendsto (λ x : ℝ, exp (b * x) / x ^ s) at_top at_top | begin
refine ((tendsto_rpow_at_top hb).comp (tendsto_exp_div_rpow_at_top (s / b))).congr' _,
filter_upwards [eventually_ge_at_top (0 : ℝ)] with x hx₀,
simp [div_rpow, (exp_pos x).le, rpow_nonneg_of_nonneg, ←rpow_mul, ←exp_mul, mul_comm x, hb.ne', *]
end | lemma | tendsto_exp_mul_div_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"exp",
"mul_comm",
"tendsto_exp_div_rpow_at_top",
"tendsto_rpow_at_top"
] | The function `exp (b * x) / x ^ s` tends to `+∞` at `+∞`, for any real `s` and `b > 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_rpow_mul_exp_neg_mul_at_top_nhds_0 (s : ℝ) (b : ℝ) (hb : 0 < b):
tendsto (λ x : ℝ, x ^ s * exp (-b * x)) at_top (𝓝 0) | begin
refine (tendsto_exp_mul_div_rpow_at_top s b hb).inv_tendsto_at_top.congr' _,
filter_upwards with x using by simp [exp_neg, inv_div, div_eq_mul_inv _ (exp _)]
end | lemma | tendsto_rpow_mul_exp_neg_mul_at_top_nhds_0 | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_eq_mul_inv",
"exp",
"exp_neg",
"inv_div",
"tendsto_exp_mul_div_rpow_at_top"
] | The function `x ^ s * exp (-b * x)` tends to `0` at `+∞`, for any real `s` and `b > 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnreal.tendsto_rpow_at_top {y : ℝ} (hy : 0 < y) :
tendsto (λ (x : ℝ≥0), x ^ y) at_top at_top | begin
rw filter.tendsto_at_top_at_top,
intros b,
obtain ⟨c, hc⟩ := tendsto_at_top_at_top.mp (tendsto_rpow_at_top hy) b,
use c.to_nnreal,
intros a ha,
exact_mod_cast hc a (real.to_nnreal_le_iff_le_coe.mp ha),
end | theorem | nnreal.tendsto_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"filter.tendsto_at_top_at_top",
"tendsto_rpow_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennreal.tendsto_rpow_at_top {y : ℝ} (hy : 0 < y) :
tendsto (λ (x : ℝ≥0∞), x ^ y) (𝓝 ⊤) (𝓝 ⊤) | begin
rw ennreal.tendsto_nhds_top_iff_nnreal,
intros x,
obtain ⟨c, _, hc⟩ :=
(at_top_basis_Ioi.tendsto_iff at_top_basis_Ioi).mp (nnreal.tendsto_rpow_at_top hy) x trivial,
have hc' : set.Ioi (↑c) ∈ 𝓝 (⊤ : ℝ≥0∞) := Ioi_mem_nhds ennreal.coe_lt_top,
refine eventually_of_mem hc' _,
intros a ha,
by_cases h... | theorem | ennreal.tendsto_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"Ioi_mem_nhds",
"ennreal.coe_lt_top",
"ennreal.coe_rpow_of_nonneg",
"ennreal.tendsto_nhds_top_iff_nnreal",
"lift",
"nnreal.tendsto_rpow_at_top",
"set.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_exp_arg_mul_im (hl : is_bounded_under (≤) l (λ x, |(g x).im|)) :
(λ x, real.exp (arg (f x) * im (g x))) =Θ[l] (λ x, (1 : ℝ)) | begin
rcases hl with ⟨b, hb⟩,
refine real.is_Theta_exp_comp_one.2 ⟨π * b, _⟩,
rw eventually_map at hb ⊢,
refine hb.mono (λ x hx, _),
erw [abs_mul],
exact mul_le_mul (abs_arg_le_pi _) hx (abs_nonneg _) real.pi_pos.le
end | lemma | complex.is_Theta_exp_arg_mul_im | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"abs_mul",
"abs_nonneg",
"mul_le_mul",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_cpow_rpow (hl : is_bounded_under (≤) l (λ x, |(g x).im|)) :
(λ x, f x ^ g x) =O[l] (λ x, abs (f x) ^ (g x).re) | calc (λ x, f x ^ g x) =O[l] (λ x, abs (f x) ^ (g x).re / real.exp (arg (f x) * im (g x))) :
is_O_of_le _ $ λ x, (abs_cpow_le _ _).trans (le_abs_self _)
... =Θ[l] (λ x, abs (f x) ^ (g x).re / (1 : ℝ)) :
(is_Theta_refl _ _).div (is_Theta_exp_arg_mul_im hl)
... =ᶠ[l] (λ x, abs (f x) ^ (g x).re) : by simp only [of_real... | lemma | complex.is_O_cpow_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_one",
"le_abs_self",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_cpow_rpow (hl_im : is_bounded_under (≤) l (λ x, |(g x).im|))
(hl : ∀ᶠ x in l, f x = 0 → re (g x) = 0 → g x = 0):
(λ x, f x ^ g x) =Θ[l] (λ x, abs (f x) ^ (g x).re) | calc (λ x, f x ^ g x) =Θ[l] (λ x, abs (f x) ^ (g x).re / real.exp (arg (f x) * im (g x))) :
is_Theta_of_norm_eventually_eq' $ hl.mono $ λ x, abs_cpow_of_imp
... =Θ[l] (λ x, abs (f x) ^ (g x).re / (1 : ℝ)) :
(is_Theta_refl _ _).div (is_Theta_exp_arg_mul_im hl_im)
... =ᶠ[l] (λ x, abs (f x) ^ (g x).re) : by simp only ... | lemma | complex.is_Theta_cpow_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_one",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_cpow_const_rpow {b : ℂ} (hl : b.re = 0 → b ≠ 0 → ∀ᶠ x in l, f x ≠ 0) :
(λ x, f x ^ b) =Θ[l] (λ x, abs (f x) ^ b.re) | is_Theta_cpow_rpow is_bounded_under_const $ by simpa only [eventually_imp_distrib_right, ne.def,
← not_frequently, not_imp_not, imp.swap] using hl | lemma | complex.is_Theta_cpow_const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"imp.swap",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.rpow (h : is_O_with c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
is_O_with (c ^ r) l (λ x, f x ^ r) (λ x, g x ^ r) | begin
apply is_O_with.of_bound,
filter_upwards [hg, h.bound] with x hgx hx,
calc |f x ^ r| ≤ |f x| ^ r : abs_rpow_le_abs_rpow _ _
... ≤ (c * |g x|) ^ r : rpow_le_rpow (abs_nonneg _) hx hr
... = c ^ r * |g x ^ r| : by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
end | lemma | asymptotics.is_O_with.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"abs_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.rpow (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) (h : f =O[l] g) :
(λ x, f x ^ r) =O[l] (λ x, g x ^ r) | let ⟨c, hc, h'⟩ := h.exists_nonneg in (h'.rpow hc hr hg).is_O | lemma | asymptotics.is_O.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.rpow (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g) :
(λ x, f x ^ r) =o[l] (λ x, g x ^ r) | is_o.of_is_O_with $ λ c hc, ((h.forall_is_O_with (rpow_pos_of_pos hc r⁻¹)).rpow
(rpow_nonneg_of_nonneg hc.le _) hr.le hg).congr_const
(by rw [←rpow_mul hc.le, inv_mul_cancel hr.ne', rpow_one]) | lemma | asymptotics.is_o.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"inv_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_rpow_exp_pos_mul_at_top (s : ℝ) {b : ℝ} (hb : 0 < b) :
(λ x : ℝ, x ^ s) =o[at_top] (λ x, exp (b * x)) | iff.mpr (is_o_iff_tendsto $ λ x h, absurd h (exp_pos _).ne') $
by simpa only [div_eq_mul_inv, exp_neg, neg_mul]
using tendsto_rpow_mul_exp_neg_mul_at_top_nhds_0 s b hb | lemma | is_o_rpow_exp_pos_mul_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_eq_mul_inv",
"exp",
"exp_neg",
"neg_mul",
"tendsto_rpow_mul_exp_neg_mul_at_top_nhds_0"
] | `x ^ s = o(exp(b * x))` as `x → ∞` for any real `s` and positive `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_zpow_exp_pos_mul_at_top (k : ℤ) {b : ℝ} (hb : 0 < b) :
(λ x : ℝ, x ^ k) =o[at_top] (λ x, exp (b * x)) | by simpa only [rpow_int_cast] using is_o_rpow_exp_pos_mul_at_top k hb | lemma | is_o_zpow_exp_pos_mul_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"exp",
"is_o_rpow_exp_pos_mul_at_top"
] | `x ^ k = o(exp(b * x))` as `x → ∞` for any integer `k` and positive `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_pow_exp_pos_mul_at_top (k : ℕ) {b : ℝ} (hb : 0 < b) :
(λ x : ℝ, x ^ k) =o[at_top] (λ x, exp (b * x)) | by simpa using is_o_zpow_exp_pos_mul_at_top k hb | lemma | is_o_pow_exp_pos_mul_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"exp",
"is_o_zpow_exp_pos_mul_at_top"
] | `x ^ k = o(exp(b * x))` as `x → ∞` for any natural `k` and positive `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_rpow_exp_at_top (s : ℝ) : (λ x : ℝ, x ^ s) =o[at_top] exp | by simpa only [one_mul] using is_o_rpow_exp_pos_mul_at_top s one_pos | lemma | is_o_rpow_exp_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"exp",
"is_o_rpow_exp_pos_mul_at_top",
"one_mul"
] | `x ^ s = o(exp x)` as `x → ∞` for any real `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_exp_neg_mul_rpow_at_top {a : ℝ} (ha : 0 < a) (b : ℝ) :
is_o at_top (λ x : ℝ, exp (-a * x)) (λ x : ℝ, x ^ b) | begin
apply is_o_of_tendsto',
{ refine (eventually_gt_at_top 0).mp (eventually_of_forall $ λ t ht h, _),
rw rpow_eq_zero_iff_of_nonneg ht.le at h,
exact (ht.ne' h.1).elim },
{ refine (tendsto_exp_mul_div_rpow_at_top (-b) a ha).inv_tendsto_at_top.congr' _,
refine (eventually_ge_at_top 0).mp (eventually... | lemma | is_o_exp_neg_mul_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"exp",
"inv_div",
"inv_inv",
"neg_mul",
"pi.inv_apply",
"real.exp_neg",
"tendsto_exp_mul_div_rpow_at_top"
] | `exp (-a * x) = o(x ^ s)` as `x → ∞`, for any positive `a` and real `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_log_rpow_at_top {r : ℝ} (hr : 0 < r) : log =o[at_top] (λ x, x ^ r) | calc log =O[at_top] (λ x, r * log x) : is_O_self_const_mul _ hr.ne' _ _
... =ᶠ[at_top] (λ x, log (x ^ r)) :
(eventually_gt_at_top 0).mono $ λ x hx, (log_rpow hx _).symm
... =o[at_top] (λ x, x ^ r) : is_o_log_id_at_top.comp_tendsto (tendsto_rpow_at_top hr) | lemma | is_o_log_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"tendsto_rpow_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_log_rpow_rpow_at_top {s : ℝ} (r : ℝ) (hs : 0 < s) :
(λ x, log x ^ r) =o[at_top] (λ x, x ^ s) | let r' := max r 1 in
have hr : 0 < r', from lt_max_iff.2 $ or.inr one_pos,
have H : 0 < s / r', from div_pos hs hr,
calc (λ x, log x ^ r) =O[at_top] (λ x, log x ^ r') :
is_O.of_bound 1 $ (tendsto_log_at_top.eventually_ge_at_top 1).mono $ λ x hx,
have hx₀ : 0 ≤ log x, from zero_le_one.trans hx,
by simp [norm_e... | lemma | is_o_log_rpow_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_mul_cancel",
"div_pos",
"is_o_log_rpow_at_top",
"le_abs_self",
"tendsto_rpow_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_abs_log_rpow_rpow_nhds_zero {s : ℝ} (r : ℝ) (hs : s < 0) :
(λ x, |log x| ^ r) =o[𝓝[>] 0] (λ x, x ^ s) | ((is_o_log_rpow_rpow_at_top r (neg_pos.2 hs)).comp_tendsto tendsto_inv_zero_at_top).congr'
(mem_of_superset (Icc_mem_nhds_within_Ioi $ set.left_mem_Ico.2 one_pos) $
λ x hx, by simp [abs_of_nonpos, log_nonpos hx.1 hx.2])
(eventually_mem_nhds_within.mono $ λ x hx,
by rw [function.comp_app, inv_rpow hx.out.le,... | lemma | is_o_abs_log_rpow_rpow_nhds_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"Icc_mem_nhds_within_Ioi",
"abs_of_nonpos",
"inv_inv",
"is_o_log_rpow_rpow_at_top",
"tendsto_inv_zero_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_log_rpow_nhds_zero {r : ℝ} (hr : r < 0) : log =o[𝓝[>] 0] (λ x, x ^ r) | (is_o_abs_log_rpow_rpow_nhds_zero 1 hr).neg_left.congr'
(mem_of_superset (Icc_mem_nhds_within_Ioi $ set.left_mem_Ico.2 one_pos) $
λ x hx, by simp [abs_of_nonpos (log_nonpos hx.1 hx.2)])
eventually_eq.rfl | lemma | is_o_log_rpow_nhds_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"Icc_mem_nhds_within_Ioi",
"abs_of_nonpos",
"is_o_abs_log_rpow_rpow_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_div_rpow_nhds_zero {r : ℝ} (hr : r < 0) :
tendsto (λ x, log x / x ^ r) (𝓝[>] 0) (𝓝 0) | (is_o_log_rpow_nhds_zero hr).tendsto_div_nhds_zero | lemma | tendsto_log_div_rpow_nhds_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"is_o_log_rpow_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_mul_rpow_nhds_zero {r : ℝ} (hr : 0 < r) :
tendsto (λ x, log x * x ^ r) (𝓝[>] 0) (𝓝 0) | (tendsto_log_div_rpow_nhds_zero $ neg_lt_zero.2 hr).congr' $
eventually_mem_nhds_within.mono $ λ x hx, by rw [rpow_neg hx.out.le, div_inv_eq_mul] | lemma | tendsto_log_mul_rpow_nhds_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_inv_eq_mul",
"tendsto_log_div_rpow_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow (x y : ℂ) : ℂ | if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) | def | complex.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"exp"
] | The complex power function `x ^ y`, given by `x ^ y = exp(y log x)` (where `log` is the
principal determination of the logarithm), unless `x = 0` where one sets `0 ^ 0 = 1` and
`0 ^ y = 0` for `y ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y | rfl | lemma | complex.cpow_eq_pow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_def (x y : ℂ) : x ^ y =
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) | rfl | lemma | complex.cpow_def | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) | if_neg hx | lemma | complex.cpow_def_of_ne_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 | by simp [cpow_def] | lemma | complex.cpow_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 | by { simp only [cpow_def], split_ifs; simp [*, exp_ne_zero] } | lemma | complex.cpow_eq_zero_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 | by simp [cpow_def, *] | lemma | complex.zero_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ (x ≠ 0 ∧ a = 0) ∨ (x = 0 ∧ a = 1) | begin
split,
{ intros hyp,
simp only [cpow_def, eq_self_iff_true, if_true] at hyp,
by_cases x = 0,
{ subst h, simp only [if_true, eq_self_iff_true] at hyp, right, exact ⟨rfl, hyp.symm⟩},
{ rw if_neg h at hyp, left, exact ⟨h, hyp.symm⟩, }, },
{ rintro (⟨h, rfl⟩|⟨rfl,rfl⟩),
{ exact zero_cpow h, ... | lemma | complex.zero_cpow_eq_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = 0 ^ x ↔ (x ≠ 0 ∧ a = 0) ∨ (x = 0 ∧ a = 1) | by rw [←zero_cpow_eq_iff, eq_comm] | lemma | complex.eq_zero_cpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_one (x : ℂ) : x ^ (1 : ℂ) = x | if hx : x = 0 then by simp [hx, cpow_def]
else by rw [cpow_def, if_neg (one_ne_zero : (1 : ℂ) ≠ 0), if_neg hx, mul_one, exp_log hx] | lemma | complex.cpow_one | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"mul_one",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 | by rw cpow_def; split_ifs; simp [one_ne_zero, *] at * | lemma | complex.one_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z | by simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole]; simp [*, exp_add, mul_add] at * | lemma | complex.cpow_add | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"boole_mul",
"exp_add",
"ite_mul",
"mul_boole",
"mul_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
x ^ (y * z) = (x ^ y) ^ z | begin
simp only [cpow_def],
split_ifs;
simp [*, exp_ne_zero, log_exp h₁ h₂, mul_assoc] at *
end | lemma | complex.cpow_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_neg (x y : ℂ) : x ^ -y = (x ^ y)⁻¹ | by simp only [cpow_def, neg_eq_zero, mul_neg]; split_ifs; simp [exp_neg] | lemma | complex.cpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"exp_neg",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z | by rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv] | lemma | complex.cpow_sub | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ | by simpa using cpow_neg x 1 | lemma | complex.cpow_neg_one | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_nat_cast (x : ℂ) : ∀ (n : ℕ), x ^ (n : ℂ) = x ^ n | | 0 := by simp
| (n + 1) := if hx : x = 0 then by simp only [hx, pow_succ,
complex.zero_cpow (nat.cast_ne_zero.2 (nat.succ_ne_zero _)), zero_mul]
else by simp [cpow_add, hx, pow_add, cpow_nat_cast n] | lemma | complex.cpow_nat_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"complex.zero_cpow",
"pow_add",
"pow_succ",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 | by { rw ← cpow_nat_cast, simp only [nat.cast_bit0, nat.cast_one] } | lemma | complex.cpow_two | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"nat.cast_bit0",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_int_cast (x : ℂ) : ∀ (n : ℤ), x ^ (n : ℂ) = x ^ n | | (n : ℕ) := by simp
| -[1+ n] := by rw zpow_neg_succ_of_nat;
simp only [int.neg_succ_of_nat_coe, int.cast_neg, complex.cpow_neg, inv_eq_one_div,
int.cast_coe_nat, cpow_nat_cast] | lemma | complex.cpow_int_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"complex.cpow_neg",
"int.cast_coe_nat",
"int.cast_neg",
"inv_eq_one_div",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x | begin
suffices : im (log x * n⁻¹) ∈ Ioc (-π) π,
{ rw [← cpow_nat_cast, ← cpow_mul _ this.1 this.2, inv_mul_cancel, cpow_one],
exact_mod_cast hn },
rw [mul_comm, ← of_real_nat_cast, ← of_real_inv, of_real_mul_im, ← div_eq_inv_mul],
rw [← pos_iff_ne_zero] at hn,
have hn' : 0 < (n : ℝ), by assumption_mod_cas... | lemma | complex.cpow_nat_inv_pow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"div_eq_inv_mul",
"div_le_iff",
"inv_mul_cancel",
"lt_div_iff",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonpos_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r | begin
rcases eq_or_ne r 0 with rfl | hr,
{ simp only [cpow_zero, mul_one] },
rcases eq_or_lt_of_le ha with rfl | ha',
{ rw [of_real_zero, zero_mul, zero_cpow hr, zero_mul] },
rcases eq_or_lt_of_le hb with rfl | hb',
{ rw [of_real_zero, mul_zero, zero_cpow hr, mul_zero] },
have ha'' : (a : ℂ) ≠ 0 := of_rea... | lemma | complex.mul_cpow_of_real_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"eq_or_lt_of_le",
"eq_or_ne",
"exp_add",
"mul_ne_zero",
"mul_one",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ | begin
simp_rw [complex.cpow_def, log_inv_eq_ite, inv_eq_zero, map_eq_zero, ite_mul, neg_mul,
is_R_or_C.conj_inv, apply_ite conj, apply_ite exp, apply_ite has_inv.inv, map_zero, map_one,
exp_neg, inv_one, inv_zero, ←exp_conj, map_mul, conj_conj],
split_ifs with hx hn ha ha; refl,
end | lemma | complex.inv_cpow_eq_ite | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"apply_ite",
"complex.cpow_def",
"exp",
"exp_neg",
"inv_eq_zero",
"inv_one",
"inv_zero",
"is_R_or_C.conj_inv",
"ite_mul",
"map_eq_zero",
"map_mul",
"map_one",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)⁻¹ | by rw [inv_cpow_eq_ite, if_neg hx] | lemma | complex.inv_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
(x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n | begin
rw [inv_cpow_eq_ite, apply_ite conj, conj_conj, conj_conj],
split_ifs,
{ refl },
{ rw inv_cpow _ _ h }
end | lemma | complex.inv_cpow_eq_ite' | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"apply_ite"
] | `complex.inv_cpow_eq_ite` with the `ite` on the other side. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_cpow_eq_ite (x : ℂ) (n : ℂ) :
conj x ^ n = if x.arg = π then x ^ n else conj (x ^ conj n) | begin
simp_rw [cpow_def, map_eq_zero, apply_ite conj, map_one, map_zero, ←exp_conj, map_mul,
conj_conj, log_conj_eq_ite],
split_ifs with hcx hn hx; refl
end | lemma | complex.conj_cpow_eq_ite | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"apply_ite",
"map_eq_zero",
"map_mul",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x ^ conj n) | by rw [conj_cpow_eq_ite, if_neg hx] | lemma | complex.conj_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) | by rw [conj_cpow _ _ hx, conj_conj] | lemma | complex.cpow_conj | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_pos (a b : ℂ) (b' : ℕ) (c : ℂ) (hb : b = b') (h : a ^ b' = c) : a ^ b = c | by rw [← h, hb, complex.cpow_nat_cast] | theorem | norm_num.cpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"complex.cpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_neg (a b : ℂ) (b' : ℕ) (c c' : ℂ)
(hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') : a ^ -b = c' | by rw [← hc, ← h, hb, complex.cpow_neg, complex.cpow_nat_cast] | theorem | norm_num.cpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"complex.cpow_nat_cast",
"complex.cpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prove_rpow' (pos neg zero : name) (α β one a b : expr) : tactic (expr × expr) | do
na ← a.to_rat,
icα ← mk_instance_cache α,
icβ ← mk_instance_cache β,
match match_sign b with
| sum.inl b := do
nc ← mk_instance_cache `(ℕ),
(icβ, nc, b', hb) ← prove_nat_uncast icβ nc b,
(icα, c, h) ← prove_pow a na icα b',
cr ← c.to_rat,
(icα, c', hc) ← prove_inv icα c cr,
pure (c'... | def | norm_num.prove_rpow' | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | Generalized version of `prove_cpow`, `prove_nnrpow`, `prove_ennrpow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prove_cpow : expr → expr → tactic (expr × expr) | prove_rpow' ``cpow_pos ``cpow_neg ``complex.cpow_zero `(ℂ) `(ℂ) `(1:ℂ) | def | norm_num.prove_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [
"complex.cpow_zero"
] | Evaluate `complex.cpow a b` where `a` is a rational numeral and `b` is an integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_cpow : expr → tactic (expr × expr) | | `(@has_pow.pow _ _ complex.has_pow %%a %%b) := b.to_int >> prove_cpow a b
| `(complex.cpow %%a %%b) := b.to_int >> prove_cpow a b
| _ := tactic.failed | def | norm_num.eval_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/complex.lean | [
"analysis.special_functions.complex.log"
] | [] | Evaluates expressions of the form `cpow a b` and `a ^ b` in the special case where
`b` is an integer and `a` is a positive rational (so it's really just a rational power). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) :
(λ (x : ℂ), (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 | begin
suffices : ∀ᶠ (x : ℂ) in (𝓝 b), x ≠ 0,
from this.mono (λ x hx, by { dsimp only, rw [zero_cpow hx, pi.zero_apply]} ),
exact is_open.eventually_mem is_open_ne hb,
end | lemma | zero_cpow_eq_nhds | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"is_open.eventually_mem",
"is_open_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_eq_nhds {a b : ℂ} (ha : a ≠ 0) :
(λ x, x ^ b) =ᶠ[𝓝 a] λ x, exp (log x * b) | begin
suffices : ∀ᶠ (x : ℂ) in (𝓝 a), x ≠ 0,
from this.mono (λ x hx, by { dsimp only, rw [cpow_def_of_ne_zero hx], }),
exact is_open.eventually_mem is_open_ne ha,
end | lemma | cpow_eq_nhds | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"exp",
"is_open.eventually_mem",
"is_open_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
(λ x, x.1 ^ x.2) =ᶠ[𝓝 p] λ x, exp (log x.1 * x.2) | begin
suffices : ∀ᶠ (x : ℂ × ℂ) in (𝓝 p), x.1 ≠ 0,
from this.mono (λ x hx, by { dsimp only, rw cpow_def_of_ne_zero hx, }),
refine is_open.eventually_mem _ hp_fst,
change is_open {x : ℂ × ℂ | x.1 = 0}ᶜ,
rw is_open_compl_iff,
exact is_closed_eq continuous_fst continuous_const,
end | lemma | cpow_eq_nhds' | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_const",
"continuous_fst",
"exp",
"is_closed_eq",
"is_open",
"is_open.eventually_mem",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_cpow {a b : ℂ} (ha : a ≠ 0) : continuous_at (λ x, a ^ x) b | begin
have cpow_eq : (λ x:ℂ, a ^ x) = λ x, exp (log a * x),
by { ext1 b, rw [cpow_def_of_ne_zero ha], },
rw cpow_eq,
exact continuous_exp.continuous_at.comp (continuous_at.mul continuous_at_const continuous_at_id),
end | lemma | continuous_at_const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.mul",
"continuous_at_const",
"continuous_at_id",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_cpow' {a b : ℂ} (h : b ≠ 0) : continuous_at (λ x, a ^ x) b | begin
by_cases ha : a = 0,
{ rw [ha, continuous_at_congr (zero_cpow_eq_nhds h)], exact continuous_at_const, },
{ exact continuous_at_const_cpow ha, },
end | lemma | continuous_at_const_cpow' | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_congr",
"continuous_at_const",
"continuous_at_const_cpow",
"zero_cpow_eq_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_cpow {p : ℂ × ℂ} (hp_fst : 0 < p.fst.re ∨ p.fst.im ≠ 0) :
continuous_at (λ x : ℂ × ℂ, x.1 ^ x.2) p | begin
have hp_fst_ne_zero : p.fst ≠ 0,
by { intro h, cases hp_fst; { rw h at hp_fst, simpa using hp_fst, }, },
rw continuous_at_congr (cpow_eq_nhds' hp_fst_ne_zero),
refine continuous_exp.continuous_at.comp _,
refine continuous_at.mul (continuous_at.comp _ continuous_fst.continuous_at)
continuous_snd.co... | lemma | continuous_at_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_at.mul",
"continuous_at_clog",
"continuous_at_congr",
"cpow_eq_nhds'"
] | The function `z ^ w` is continuous in `(z, w)` provided that `z` does not belong to the interval
`(-∞, 0]` on the real line. See also `complex.continuous_at_cpow_zero_of_re_pos` for a version that
works for `z = 0` but assumes `0 < re w`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_cpow_const {a b : ℂ} (ha : 0 < a.re ∨ a.im ≠ 0) :
continuous_at (λ x, cpow x b) a | tendsto.comp (@continuous_at_cpow (a, b) ha) (continuous_at_id.prod continuous_at_const) | lemma | continuous_at_cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_const",
"continuous_at_cpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cpow {l : filter α} {f g : α → ℂ} {a b : ℂ} (hf : tendsto f l (𝓝 a))
(hg : tendsto g l (𝓝 b)) (ha : 0 < a.re ∨ a.im ≠ 0) :
tendsto (λ x, f x ^ g x) l (𝓝 (a ^ b)) | (@continuous_at_cpow (a,b) ha).tendsto.comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at_cpow",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.const_cpow {l : filter α} {f : α → ℂ} {a b : ℂ} (hf : tendsto f l (𝓝 b))
(h : a ≠ 0 ∨ b ≠ 0) :
tendsto (λ x, a ^ f x) l (𝓝 (a ^ b)) | begin
cases h,
{ exact (continuous_at_const_cpow h).tendsto.comp hf, },
{ exact (continuous_at_const_cpow' h).tendsto.comp hf, },
end | lemma | filter.tendsto.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at_const_cpow",
"continuous_at_const_cpow'",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.cpow (hf : continuous_within_at f s a) (hg : continuous_within_at g s a)
(h0 : 0 < (f a).re ∨ (f a).im ≠ 0) :
continuous_within_at (λ x, f x ^ g x) s a | hf.cpow hg h0 | lemma | continuous_within_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.const_cpow {b : ℂ} (hf : continuous_within_at f s a)
(h : b ≠ 0 ∨ f a ≠ 0) :
continuous_within_at (λ x, b ^ f x) s a | hf.const_cpow h | lemma | continuous_within_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.cpow (hf : continuous_at f a) (hg : continuous_at g a)
(h0 : 0 < (f a).re ∨ (f a).im ≠ 0) :
continuous_at (λ x, f x ^ g x) a | hf.cpow hg h0 | lemma | continuous_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.const_cpow {b : ℂ} (hf : continuous_at f a) (h : b ≠ 0 ∨ f a ≠ 0) :
continuous_at (λ x, b ^ f x) a | hf.const_cpow h | lemma | continuous_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.cpow (hf : continuous_on f s) (hg : continuous_on g s)
(h0 : ∀ a ∈ s, 0 < (f a).re ∨ (f a).im ≠ 0) :
continuous_on (λ x, f x ^ g x) s | λ a ha, (hf a ha).cpow (hg a ha) (h0 a ha) | lemma | continuous_on.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.const_cpow {b : ℂ} (hf : continuous_on f s) (h : b ≠ 0 ∨ ∀ a ∈ s, f a ≠ 0) :
continuous_on (λ x, b ^ f x) s | λ a ha, (hf a ha).const_cpow (h.imp id $ λ h, h a ha) | lemma | continuous_on.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.cpow (hf : continuous f) (hg : continuous g)
(h0 : ∀ a, 0 < (f a).re ∨ (f a).im ≠ 0) :
continuous (λ x, f x ^ g x) | continuous_iff_continuous_at.2 $ λ a, (hf.continuous_at.cpow hg.continuous_at (h0 a)) | lemma | continuous.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.const_cpow {b : ℂ} (hf : continuous f) (h : b ≠ 0 ∨ ∀ a, f a ≠ 0) :
continuous (λ x, b ^ f x) | continuous_iff_continuous_at.2 $ λ a, (hf.continuous_at.const_cpow $ h.imp id $ λ h, h a) | lemma | continuous.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.cpow_const {b : ℂ} (hf : continuous_on f s)
(h : ∀ (a : α), a ∈ s → 0 < (f a).re ∨ (f a).im ≠ 0) :
continuous_on (λ x, (f x) ^ b) s | hf.cpow continuous_on_const h | lemma | continuous_on.cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_on",
"continuous_on_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_rpow {a b : ℝ} (h : a ≠ 0) : continuous_at (rpow a) b | begin
have : rpow a = λ x : ℝ, ((a : ℂ) ^ (x : ℂ)).re, by { ext1 x, rw [rpow_eq_pow, rpow_def], },
rw this,
refine complex.continuous_re.continuous_at.comp _,
refine (continuous_at_const_cpow _).comp complex.continuous_of_real.continuous_at,
norm_cast,
exact h,
end | lemma | real.continuous_at_const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_const_cpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_rpow' {a b : ℝ} (h : b ≠ 0) : continuous_at (rpow a) b | begin
have : rpow a = λ x : ℝ, ((a : ℂ) ^ (x : ℂ)).re, by { ext1 x, rw [rpow_eq_pow, rpow_def], },
rw this,
refine complex.continuous_re.continuous_at.comp _,
refine (continuous_at_const_cpow' _).comp complex.continuous_of_real.continuous_at,
norm_cast,
exact h,
end | lemma | real.continuous_at_const_rpow' | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_const_cpow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_nhds_of_neg {p : ℝ × ℝ} (hp_fst : p.fst < 0) :
(λ x : ℝ × ℝ, x.1 ^ x.2) =ᶠ[𝓝 p] λ x, exp (log x.1 * x.2) * cos (x.2 * π) | begin
suffices : ∀ᶠ (x : ℝ × ℝ) in (𝓝 p), x.1 < 0,
from this.mono (λ x hx, by { dsimp only, rw rpow_def_of_neg hx, }),
exact is_open.eventually_mem (is_open_lt continuous_fst continuous_const) hp_fst,
end | lemma | real.rpow_eq_nhds_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_const",
"continuous_fst",
"exp",
"is_open.eventually_mem",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_nhds_of_pos {p : ℝ × ℝ} (hp_fst : 0 < p.fst) :
(λ x : ℝ × ℝ, x.1 ^ x.2) =ᶠ[𝓝 p] λ x, exp (log x.1 * x.2) | begin
suffices : ∀ᶠ (x : ℝ × ℝ) in (𝓝 p), 0 < x.1,
from this.mono (λ x hx, by { dsimp only, rw rpow_def_of_pos hx, }),
exact is_open.eventually_mem (is_open_lt continuous_const continuous_fst) hp_fst,
end | lemma | real.rpow_eq_nhds_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_const",
"continuous_fst",
"exp",
"is_open.eventually_mem",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
continuous_at (λ p : ℝ × ℝ, p.1 ^ p.2) p | begin
rw ne_iff_lt_or_gt at hp,
cases hp,
{ rw continuous_at_congr (rpow_eq_nhds_of_neg hp),
refine continuous_at.mul _ (continuous_cos.continuous_at.comp _),
{ refine continuous_exp.continuous_at.comp (continuous_at.mul _ continuous_snd.continuous_at),
refine (continuous_at_log _).comp continuous_f... | lemma | real.continuous_at_rpow_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.mul",
"continuous_at_congr",
"continuous_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.2) :
continuous_at (λ p : ℝ × ℝ, p.1 ^ p.2) p | begin
cases p with x y,
obtain hx|rfl := ne_or_eq x 0,
{ exact continuous_at_rpow_of_ne (x, y) hx },
have A : tendsto (λ p : ℝ × ℝ, exp (log p.1 * p.2)) (𝓝[≠] 0 ×ᶠ 𝓝 y) (𝓝 0) :=
tendsto_exp_at_bot.comp
((tendsto_log_nhds_within_zero.comp tendsto_fst).at_bot_mul hp tendsto_snd),
have B : tendsto (... | lemma | real.continuous_at_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"exp",
"lt_mem_nhds",
"ne_or_eq",
"nhds_prod_eq",
"nhds_within_singleton",
"nhds_within_union",
"nhds_within_univ",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow (p : ℝ × ℝ) (h : p.1 ≠ 0 ∨ 0 < p.2) :
continuous_at (λ p : ℝ × ℝ, p.1 ^ p.2) p | h.elim (λ h, continuous_at_rpow_of_ne p h) (λ h, continuous_at_rpow_of_pos p h) | lemma | real.continuous_at_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow_const (x : ℝ) (q : ℝ) (h : x ≠ 0 ∨ 0 < q) :
continuous_at (λ (x : ℝ), x ^ q) x | begin
change continuous_at ((λ p : ℝ × ℝ, p.1 ^ p.2) ∘ (λ y : ℝ, (y, q))) x,
apply continuous_at.comp,
{ exact continuous_at_rpow (x, q) h },
{ exact (continuous_id'.prod_mk continuous_const).continuous_at }
end | lemma | real.continuous_at_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.rpow {l : filter α} {f g : α → ℝ} {x y : ℝ}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) (h : x ≠ 0 ∨ 0 < y) :
tendsto (λ t, f t ^ g t) l (𝓝 (x ^ y)) | (real.continuous_at_rpow (x, y) h).tendsto.comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"filter",
"real.continuous_at_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.rpow_const {l : filter α} {f : α → ℝ} {x p : ℝ}
(hf : tendsto f l (𝓝 x)) (h : x ≠ 0 ∨ 0 ≤ p) :
tendsto (λ a, f a ^ p) l (𝓝 (x ^ p)) | if h0 : 0 = p then h0 ▸ by simp [tendsto_const_nhds]
else hf.rpow tendsto_const_nhds (h.imp id $ λ h', h'.lt_of_ne h0) | lemma | filter.tendsto.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.rpow (hf : continuous_at f x) (hg : continuous_at g x) (h : f x ≠ 0 ∨ 0 < g x) :
continuous_at (λ t, f t ^ g t) x | hf.rpow hg h | lemma | continuous_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.rpow (hf : continuous_within_at f s x) (hg : continuous_within_at g s x)
(h : f x ≠ 0 ∨ 0 < g x) :
continuous_within_at (λ t, f t ^ g t) s x | hf.rpow hg h | lemma | continuous_within_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.rpow (hf : continuous_on f s) (hg : continuous_on g s)
(h : ∀ x ∈ s, f x ≠ 0 ∨ 0 < g x) :
continuous_on (λ t, f t ^ g t) s | λ t ht, (hf t ht).rpow (hg t ht) (h t ht) | lemma | continuous_on.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.rpow (hf : continuous f) (hg : continuous g) (h : ∀ x, f x ≠ 0 ∨ 0 < g x) :
continuous (λ x, f x ^ g x) | continuous_iff_continuous_at.2 $ λ x, (hf.continuous_at.rpow hg.continuous_at (h x)) | lemma | continuous.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.rpow_const (hf : continuous_within_at f s x) (h : f x ≠ 0 ∨ 0 ≤ p) :
continuous_within_at (λ x, f x ^ p) s x | hf.rpow_const h | lemma | continuous_within_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.rpow_const (hf : continuous_at f x) (h : f x ≠ 0 ∨ 0 ≤ p) :
continuous_at (λ x, f x ^ p) x | hf.rpow_const h | lemma | continuous_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.rpow_const (hf : continuous_on f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 0 ≤ p) :
continuous_on (λ x, f x ^ p) s | λ x hx, (hf x hx).rpow_const (h x hx) | lemma | continuous_on.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.rpow_const (hf : continuous f) (h : ∀ x, f x ≠ 0 ∨ 0 ≤ p) :
continuous (λ x, f x ^ p) | continuous_iff_continuous_at.2 $ λ x, hf.continuous_at.rpow_const (h x) | lemma | continuous.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_cpow_zero_of_re_pos {z : ℂ} (hz : 0 < z.re) :
continuous_at (λ x : ℂ × ℂ, x.1 ^ x.2) (0, z) | begin
have hz₀ : z ≠ 0, from ne_of_apply_ne re hz.ne',
rw [continuous_at, zero_cpow hz₀, tendsto_zero_iff_norm_tendsto_zero],
refine squeeze_zero (λ _, norm_nonneg _) (λ _, abs_cpow_le _ _) _,
simp only [div_eq_mul_inv, ← real.exp_neg],
refine tendsto.zero_mul_is_bounded_under_le _ _,
{ convert (continuous_... | lemma | complex.continuous_at_cpow_zero_of_re_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"abs_of_pos",
"continuous_at",
"continuous_snd",
"div_eq_mul_inv",
"gt_mem_nhds",
"mul_le_mul",
"ne_of_apply_ne",
"neg_le_abs_self",
"real.exp_neg",
"real.exp_pos",
"real.norm_eq_abs",
"real.zero_rpow",
"squeeze_zero"
] | See also `continuous_at_cpow` and `complex.continuous_at_cpow_of_re_pos`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_cpow_of_re_pos {p : ℂ × ℂ} (h₁ : 0 ≤ p.1.re ∨ p.1.im ≠ 0) (h₂ : 0 < p.2.re) :
continuous_at (λ x : ℂ × ℂ, x.1 ^ x.2) p | begin
cases p with z w,
rw [← not_lt_zero_iff, lt_iff_le_and_ne, not_and_distrib, ne.def, not_not, not_le_zero_iff] at h₁,
rcases h₁ with h₁|(rfl : z = 0),
exacts [continuous_at_cpow h₁, continuous_at_cpow_zero_of_re_pos h₂]
end | lemma | complex.continuous_at_cpow_of_re_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_cpow",
"lt_iff_le_and_ne",
"not_and_distrib",
"not_not"
] | See also `continuous_at_cpow` for a version that assumes `p.1 ≠ 0` but makes no
assumptions about `p.2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_cpow_const_of_re_pos {z w : ℂ} (hz : 0 ≤ re z ∨ im z ≠ 0) (hw : 0 < re w) :
continuous_at (λ x, x ^ w) z | tendsto.comp (@continuous_at_cpow_of_re_pos (z, w) hz hw)
(continuous_at_id.prod continuous_at_const) | lemma | complex.continuous_at_cpow_const_of_re_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_const"
] | See also `continuous_at_cpow_const` for a version that assumes `z ≠ 0` but makes no
assumptions about `w`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_real_cpow (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) :
continuous_at (λ p, ↑p.1 ^ p.2 : ℝ × ℂ → ℂ) (x, y) | begin
rcases lt_trichotomy 0 x with hx | rfl | hx,
{ -- x > 0 : easy case
have : continuous_at (λ p, ⟨↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y),
from continuous_of_real.continuous_at.prod_map continuous_at_id,
refine (continuous_at_cpow (or.inl _)).comp this,
rwa of_real_re },
{ -- x = 0 : reduce to co... | lemma | complex.continuous_at_of_real_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"Iio_mem_nhds",
"continuous_at",
"continuous_at.comp",
"continuous_at.mul",
"continuous_at.prod_map",
"continuous_at_cpow",
"continuous_at_id",
"continuous_snd",
"exp",
"prod_mem_nhds"
] | Continuity of `(x, y) ↦ x ^ y` as a function on `ℝ × ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_real_cpow_const (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) :
continuous_at (λ a, a ^ y : ℝ → ℂ) x | @continuous_at.comp _ _ _ _ _ _ _ _ x (continuous_at_of_real_cpow x y h)
(continuous_id.prod_mk continuous_const).continuous_at | lemma | complex.continuous_at_of_real_cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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