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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
has_deriv_within_at.arsinh (hf : has_deriv_within_at f f' s a) :
has_deriv_within_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') s a | (has_deriv_at_arsinh _).comp_has_deriv_within_at a hf | lemma | has_deriv_within_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bernstein (n ν : ℕ) : C(I, ℝ) | (bernstein_polynomial ℝ n ν).to_continuous_map_on I | def | bernstein | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"bernstein_polynomial"
] | The Bernstein polynomials, as continuous functions on `[0,1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bernstein_apply (n ν : ℕ) (x : I) :
bernstein n ν x = n.choose ν * x^ν * (1-x)^(n-ν) | begin
dsimp [bernstein, polynomial.to_continuous_map_on, polynomial.to_continuous_map,
bernstein_polynomial],
simp,
end | lemma | bernstein_apply | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"bernstein",
"bernstein_polynomial",
"polynomial.to_continuous_map",
"polynomial.to_continuous_map_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bernstein_nonneg {n ν : ℕ} {x : I} :
0 ≤ bernstein n ν x | begin
simp only [bernstein_apply],
exact mul_nonneg
(mul_nonneg (nat.cast_nonneg _) (pow_nonneg (by unit_interval) _))
(pow_nonneg (by unit_interval) _),
end | lemma | bernstein_nonneg | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"bernstein",
"bernstein_apply",
"nat.cast_nonneg",
"pow_nonneg",
"unit_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
z {n : ℕ} (k : fin (n+1)) : I | ⟨(k : ℝ) / n,
begin
cases n,
{ norm_num },
{ have h₁ : 0 < (n.succ : ℝ) := by exact_mod_cast (nat.succ_pos _),
have h₂ : ↑k ≤ n.succ := by exact_mod_cast (fin.le_last k),
rw [set.mem_Icc, le_div_iff h₁, div_le_iff h₁],
norm_cast,
simp [h₂], },
end⟩ | def | bernstein.z | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"div_le_iff",
"fin.le_last",
"le_div_iff",
"set.mem_Icc"
] | Send `k : fin (n+1)` to the equally spaced points `k/n` in the unit interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
probability (n : ℕ) (x : I) :
∑ k : fin (n+1), bernstein n k x = 1 | begin
have := bernstein_polynomial.sum ℝ n,
apply_fun (λ p, polynomial.aeval (x : ℝ) p) at this,
simp [alg_hom.map_sum, finset.sum_range] at this,
exact this,
end | lemma | bernstein.probability | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"alg_hom.map_sum",
"bernstein",
"bernstein_polynomial.sum",
"polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
∑ k : fin (n+1), (x - k/ₙ : ℝ)^2 * bernstein n k x = x * (1-x) / n | begin
have h' : (n : ℝ) ≠ 0 := ne_of_gt h,
apply_fun (λ x : ℝ, x * n) using group_with_zero.mul_right_injective h',
apply_fun (λ x : ℝ, x * n) using group_with_zero.mul_right_injective h',
dsimp,
conv_lhs { simp only [finset.sum_mul, z], },
conv_rhs { rw div_mul_cancel _ h', },
have := bernstein_polynomia... | lemma | bernstein.variance | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"alg_hom.map_sum",
"bernstein",
"bernstein_polynomial.variance",
"div_mul_cancel",
"finset.sum_mul",
"group_with_zero.mul_right_injective",
"mul_comm",
"polynomial.aeval",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bernstein_approximation (n : ℕ) (f : C(I, ℝ)) : C(I, ℝ) | ∑ k : fin (n+1), f k/ₙ • bernstein n k | def | bernstein_approximation | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"bernstein"
] | The `n`-th approximation of a continuous function on `[0,1]` by Bernstein polynomials,
given by `∑ k, f (k/n) * bernstein n k x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply (n : ℕ) (f : C(I, ℝ)) (x : I) :
bernstein_approximation n f x = ∑ k : fin (n+1), f k/ₙ * bernstein n k x | by simp [bernstein_approximation] | lemma | bernstein_approximation.apply | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"bernstein",
"bernstein_approximation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ (f : C(I, ℝ)) (ε : ℝ) (h : 0 < ε) : ℝ | f.modulus (ε/2) (half_pos h) | def | bernstein_approximation.δ | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"half_pos"
] | The modulus of (uniform) continuity for `f`, chosen so `|f x - f y| < ε/2` when `|x - y| < δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_pos {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} : 0 < δ f ε h | f.modulus_pos | lemma | bernstein_approximation.δ_pos | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
S (f : C(I, ℝ)) (ε : ℝ) (h : 0 < ε) (n : ℕ) (x : I) : finset (fin (n+1)) | { k : fin (n+1) | dist k/ₙ x < δ f ε h }.to_finset | def | bernstein_approximation.S | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"finset"
] | The set of points `k` so `k/n` is within `δ` of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_of_mem_S
{f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : fin (n+1)} (m : k ∈ S f ε h n x) : | |f k/ₙ - f x| < ε/2 :=
begin
apply f.dist_lt_of_dist_lt_modulus (ε/2) (half_pos h),
simpa [S] using m,
end | lemma | bernstein_approximation.lt_of_mem_S | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"half_pos"
] | If `k ∈ S`, then `f(k/n)` is close to `f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_mem_S_compl
{f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : fin (n+1)} (m : k ∈ (S f ε h n x)ᶜ) :
(1 : ℝ) ≤ (δ f ε h)^(-2 : ℤ) * (x - k/ₙ) ^ 2 | begin
simp only [finset.mem_compl, not_lt, set.mem_to_finset, set.mem_set_of_eq, S] at m,
rw [zpow_neg, ← div_eq_inv_mul, zpow_two, ←pow_two, one_le_div (pow_pos δ_pos 2), sq_le_sq,
abs_of_pos δ_pos],
rwa [dist_comm] at m
end | lemma | bernstein_approximation.le_of_mem_S_compl | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"abs_of_pos",
"dist_comm",
"div_eq_inv_mul",
"finset.mem_compl",
"one_le_div",
"pow_pos",
"set.mem_to_finset",
"sq_le_sq",
"zpow_neg",
"zpow_two"
] | If `k ∉ S`, then as `δ ≤ |x - k/n|`, we have the inequality `1 ≤ δ^-2 * (x - k/n)^2`.
This particular formulation will be helpful later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bernstein_approximation_uniform (f : C(I, ℝ)) :
tendsto (λ n : ℕ, bernstein_approximation n f) at_top (𝓝 f) | begin
simp only [metric.nhds_basis_ball.tendsto_right_iff, metric.mem_ball, dist_eq_norm],
intros ε h,
let δ := δ f ε h,
have nhds_zero := tendsto_const_div_at_top_nhds_0_nat (2 * ‖f‖ * δ ^ (-2 : ℤ)),
filter_upwards [nhds_zero.eventually (gt_mem_nhds (half_pos h)), eventually_gt_at_top 0]
with n nh npos',... | theorem | bernstein_approximation_uniform | analysis.special_functions | src/analysis/special_functions/bernstein.lean | [
"analysis.specific_limits.basic",
"ring_theory.polynomial.bernstein",
"topology.continuous_function.polynomial",
"topology.continuous_function.compact"
] | [
"abs_mul",
"add_halves",
"bernstein",
"bernstein.probability",
"bernstein_approximation",
"bernstein_nonneg",
"continuous_map.norm_lt_iff",
"div_le_div_right",
"finset.abs_sum_le_sum_abs",
"finset.mul_sum",
"gt_mem_nhds",
"half_pos",
"metric.mem_ball",
"mul_assoc",
"mul_le_mul_of_nonneg_... | The Bernstein approximations
```
∑ k : fin (n+1), f (k/n : ℝ) * n.choose k * x^k * (1-x)^(n-k)
```
for a continuous function `f : C([0,1], ℝ)` converge uniformly to `f` as `n` tends to infinity.
This is the proof given in [Richard Beals' *Analysis, an introduction*][beals-analysis], §7D,
and reproduced on wikipedia. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_exp_cmp_filter (l : filter ℂ) : Prop | (tendsto_re : tendsto re l at_top)
(is_O_im_pow_re : ∀ n : ℕ, (λ z : ℂ, z.im ^ n) =O[l] (λ z, real.exp z.re)) | structure | complex.is_exp_cmp_filter | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"filter",
"real.exp"
] | We say that `l : filter ℂ` is an *exponential comparison filter* if the real part tends to
infinity along `l` and the imaginary part grows subexponentially compared to the real part. These
properties guarantee that `(λ z, z ^ a₁ * exp (b₁ * z)) =o[l] (λ z, z ^ a₂ * exp (b₂ * z))` for any
complex `a₁`, `a₂` and real `b₁... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_O_im_re_rpow (hre : tendsto re l at_top) (r : ℝ) (hr : im =O[l] (λ z, z.re ^ r)) :
is_exp_cmp_filter l | ⟨hre, λ n, is_o.is_O $
calc (λ z : ℂ, z.im ^ n) =O[l] (λ z, (z.re ^ r) ^ n) : hr.pow n
... =ᶠ[l] (λ z, z.re ^ (r * n)) : (hre.eventually_ge_at_top 0).mono $
λ z hz, by simp only [real.rpow_mul hz r n, real.rpow_nat_cast]
... =o[l] (λ z, real.exp z.re) : (is_o_rpow_exp_at_top _).comp_tendsto hre⟩ | lemma | complex.is_exp_cmp_filter.of_is_O_im_re_rpow | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"is_o_rpow_exp_at_top",
"real.exp",
"real.rpow_mul",
"real.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_O_im_re_pow (hre : tendsto re l at_top) (n : ℕ) (hr : im =O[l] (λ z, z.re ^ n)) :
is_exp_cmp_filter l | of_is_O_im_re_rpow hre n $ by simpa only [real.rpow_nat_cast] | lemma | complex.is_exp_cmp_filter.of_is_O_im_re_pow | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"real.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bounded_under_abs_im (hre : tendsto re l at_top)
(him : is_bounded_under (≤) l (λ z, |z.im|)) :
is_exp_cmp_filter l | of_is_O_im_re_pow hre 0 $
by simpa only [pow_zero] using @is_bounded_under.is_O_const ℂ ℝ ℝ _ _ _ l him 1 one_ne_zero | lemma | complex.is_exp_cmp_filter.of_bounded_under_abs_im | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"one_ne_zero",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bounded_under_im (hre : tendsto re l at_top) (him_le : is_bounded_under (≤) l im)
(him_ge : is_bounded_under (≥) l im) :
is_exp_cmp_filter l | of_bounded_under_abs_im hre $ is_bounded_under_le_abs.2 ⟨him_le, him_ge⟩ | lemma | complex.is_exp_cmp_filter.of_bounded_under_im | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_ne (hl : is_exp_cmp_filter l) : ∀ᶠ w : ℂ in l, w ≠ 0 | hl.tendsto_re.eventually_ne_at_top' _ | lemma | complex.is_exp_cmp_filter.eventually_ne | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs_re (hl : is_exp_cmp_filter l) : tendsto (λ z : ℂ, |z.re|) l at_top | tendsto_abs_at_top_at_top.comp hl.tendsto_re | lemma | complex.is_exp_cmp_filter.tendsto_abs_re | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs (hl : is_exp_cmp_filter l) : tendsto abs l at_top | tendsto_at_top_mono abs_re_le_abs hl.tendsto_abs_re | lemma | complex.is_exp_cmp_filter.tendsto_abs | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_log_re_re (hl : is_exp_cmp_filter l) : (λ z, real.log z.re) =o[l] re | real.is_o_log_id_at_top.comp_tendsto hl.tendsto_re | lemma | complex.is_exp_cmp_filter.is_o_log_re_re | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"real.log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_im_pow_exp_re (hl : is_exp_cmp_filter l) (n : ℕ) :
(λ z : ℂ, z.im ^ n) =o[l] (λ z, real.exp z.re) | flip is_o.of_pow two_ne_zero $
calc (λ z : ℂ, (z.im ^ n) ^ 2) = (λ z, z.im ^ (2 * n)) : by simp only [pow_mul']
... =O[l] (λ z, real.exp z.re) : hl.is_O_im_pow_re _
... = (λ z, (real.exp z.re) ^ 1) : by simp only [pow_one]
... =o[l] (λ z, (real.exp z.re) ^ 2) : (is_o_pow_pow_at_top_of_lt one_lt_two).comp_te... | lemma | complex.is_exp_cmp_filter.is_o_im_pow_exp_re | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"one_lt_two",
"pow_mul'",
"pow_one",
"real.exp",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_im_pow_eventually_le_exp_re (hl : is_exp_cmp_filter l) (n : ℕ) :
(λ z : ℂ, |z.im| ^ n) ≤ᶠ[l] (λ z, real.exp z.re) | by simpa using (hl.is_o_im_pow_exp_re n).bound zero_lt_one | lemma | complex.is_exp_cmp_filter.abs_im_pow_eventually_le_exp_re | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"bound",
"real.exp",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_log_abs_re (hl : is_exp_cmp_filter l) : (λ z, real.log (abs z)) =o[l] re | calc (λ z, real.log (abs z)) =O[l] (λ z, real.log (real.sqrt 2) + real.log (max z.re (|z.im|))) :
is_O.of_bound 1 $ (hl.tendsto_re.eventually_ge_at_top 1).mono $ λ z hz,
begin
have h2 : 0 < real.sqrt 2, by simp,
have hz' : 1 ≤ abs z, from hz.trans (re_le_abs z),
have hz₀ : 0 < abs z, from one_po... | lemma | complex.is_exp_cmp_filter.is_o_log_abs_re | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"abs_of_pos",
"le_abs_self",
"one_mul",
"pow_pos",
"real.log",
"real.log_le_iff_le_exp",
"real.log_le_log",
"real.log_mul",
"real.log_nonneg",
"real.log_pos",
"real.log_pow",
"real.norm_eq_abs",
"real.sqrt",
"zero_le_one"
] | If `l : filter ℂ` is an "exponential comparison filter", then $\log |z| =o(ℜ z)$ along `l`.
This is the main lemma in the proof of `complex.is_exp_cmp_filter.is_o_cpow_exp` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_cpow_exp (hl : is_exp_cmp_filter l) (a : ℂ) {b : ℝ} (hb : 0 < b) :
(λ z, z ^ a) =o[l] (λ z, exp (b * z)) | calc (λ z, z ^ a) =Θ[l] λ z, abs z ^ re a : is_Theta_cpow_const_rpow $ λ _ _, hl.eventually_ne
... =ᶠ[l] λ z, real.exp (re a * real.log (abs z)) : hl.eventually_ne.mono $
λ z hz, by simp only [real.rpow_def_of_pos, abs.pos hz, mul_comm]
... =o[l] λ z, exp (b * z) : is_o.of_norm_right $
begin
simp only [norm_eq_... | lemma | complex.is_exp_cmp_filter.is_o_cpow_exp | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"exp",
"mul_comm",
"real.exp",
"real.is_o_exp_comp_exp_comp",
"real.log",
"real.rpow_def_of_pos"
] | If `l : filter ℂ` is an "exponential comparison filter", then for any complex `a` and any
positive real `b`, we have `(λ z, z ^ a) =o[l] (λ z, exp (b * z))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_cpow_mul_exp {b₁ b₂ : ℝ} (hl : is_exp_cmp_filter l) (hb : b₁ < b₂) (a₁ a₂ : ℂ) :
(λ z, z ^ a₁ * exp (b₁ * z)) =o[l] (λ z, z ^ a₂ * exp (b₂ * z)) | calc (λ z, z ^ a₁ * exp (b₁ * z)) =ᶠ[l] (λ z, z ^ a₂ * exp (b₁ * z) * z ^ (a₁ - a₂)) :
hl.eventually_ne.mono $ λ z hz,
by { simp only, rw [mul_right_comm, ← cpow_add _ _ hz, add_sub_cancel'_right] }
... =o[l] λ z, z ^ a₂ * exp (b₁ * z) * exp (↑(b₂ - b₁) * z) :
(is_O_refl (λ z, z ^ a₂ * exp (b₁ * z)) l).mul_is_o... | lemma | complex.is_exp_cmp_filter.is_o_cpow_mul_exp | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"exp",
"exp_add",
"mul_assoc",
"mul_right_comm"
] | If `l : filter ℂ` is an "exponential comparison filter", then for any complex `a₁`, `a₂` and any
real `b₁ < b₂`, we have `(λ z, z ^ a₁ * exp (b₁ * z)) =o[l] (λ z, z ^ a₂ * exp (b₂ * z))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_exp_cpow (hl : is_exp_cmp_filter l) (a : ℂ) {b : ℝ} (hb : b < 0) :
(λ z, exp (b * z)) =o[l] (λ z, z ^ a) | by simpa using hl.is_o_cpow_mul_exp hb 0 a | lemma | complex.is_exp_cmp_filter.is_o_exp_cpow | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"exp"
] | If `l : filter ℂ` is an "exponential comparison filter", then for any complex `a` and any
negative real `b`, we have `(λ z, exp (b * z)) =o[l] (λ z, z ^ a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_pow_mul_exp {b₁ b₂ : ℝ} (hl : is_exp_cmp_filter l) (hb : b₁ < b₂) (m n : ℕ) :
(λ z, z ^ m * exp (b₁ * z)) =o[l] (λ z, z ^ n * exp (b₂ * z)) | by simpa only [cpow_nat_cast] using hl.is_o_cpow_mul_exp hb m n | lemma | complex.is_exp_cmp_filter.is_o_pow_mul_exp | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"exp"
] | If `l : filter ℂ` is an "exponential comparison filter", then for any complex `a₁`, `a₂` and any
natural `b₁ < b₂`, we have `(λ z, z ^ a₁ * exp (b₁ * z)) =o[l] (λ z, z ^ a₂ * exp (b₂ * z))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_zpow_mul_exp {b₁ b₂ : ℝ} (hl : is_exp_cmp_filter l) (hb : b₁ < b₂) (m n : ℤ) :
(λ z, z ^ m * exp (b₁ * z)) =o[l] (λ z, z ^ n * exp (b₂ * z)) | by simpa only [cpow_int_cast] using hl.is_o_cpow_mul_exp hb m n | lemma | complex.is_exp_cmp_filter.is_o_zpow_mul_exp | analysis.special_functions | src/analysis/special_functions/compare_exp.lean | [
"analysis.special_functions.pow.asymptotics",
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics"
] | [
"exp"
] | If `l : filter ℂ` is an "exponential comparison filter", then for any complex `a₁`, `a₂` and any
integer `b₁ < b₂`, we have `(λ z, z ^ a₁ * exp (b₁ * z)) =o[l] (λ z, z ^ a₂ * exp (b₂ * z))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_bound_sq (x z : ℂ) (hz : ‖z‖ ≤ 1) :
‖exp (x + z) - exp x - z • exp x‖ ≤ ‖exp x‖ * ‖z‖ ^ 2 | calc ‖exp (x + z) - exp x - z * exp x‖
= ‖exp x * (exp z - 1 - z)‖ : by { congr, rw [exp_add], ring }
... = ‖exp x‖ * ‖exp z - 1 - z‖ : norm_mul _ _
... ≤ ‖exp x‖ * ‖z‖^2 : mul_le_mul_of_nonneg_left (abs_exp_sub_one_sub_id_le hz) (norm_nonneg _) | lemma | complex.exp_bound_sq | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"exp_add",
"mul_le_mul_of_nonneg_left",
"norm_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_lipschitz_exp {r : ℝ} (hr_nonneg : 0 ≤ r) (hr_le : r ≤ 1) (x y : ℂ)
(hyx : ‖y - x‖ < r) :
‖exp y - exp x‖ ≤ (1 + r) * ‖exp x‖ * ‖y - x‖ | begin
have hy_eq : y = x + (y - x), by abel,
have hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖,
{ rw pow_two,
exact mul_le_mul hyx.le le_rfl (norm_nonneg _) hr_nonneg, },
have h_sq : ∀ z, ‖z‖ ≤ 1 → ‖exp (x + z) - exp x‖ ≤ ‖z‖ * ‖exp x‖ + ‖exp x‖ * ‖z‖ ^ 2,
{ intros z hz,
have : ‖exp (x + z) - exp x - z • exp... | lemma | complex.locally_lipschitz_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"le_rfl",
"mul_le_mul",
"norm_smul",
"pow_two",
"ring",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_exp : continuous exp | continuous_iff_continuous_at.mpr $
λ x, continuous_at_of_locally_lipschitz zero_lt_one (2 * ‖exp x‖)
(locally_lipschitz_exp zero_le_one le_rfl x) | lemma | complex.continuous_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous",
"continuous_at_of_locally_lipschitz",
"exp",
"le_rfl",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_exp {s : set ℂ} : continuous_on exp s | continuous_exp.continuous_on | lemma | complex.continuous_on_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_on",
"continuous_on_exp",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cexp {l : filter α} {f : α → ℂ} {z : ℂ} (hf : tendsto f l (𝓝 z)) :
tendsto (λ x, exp (f x)) l (𝓝 (exp z)) | (continuous_exp.tendsto _).comp hf | lemma | filter.tendsto.cexp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.cexp (h : continuous_within_at f s x) :
continuous_within_at (λ y, exp (f y)) s x | h.cexp | lemma | continuous_within_at.cexp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_within_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.cexp (h : continuous_at f x) : continuous_at (λ y, exp (f y)) x | h.cexp | lemma | continuous_at.cexp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.cexp (h : continuous_on f s) : continuous_on (λ y, exp (f y)) s | λ x hx, (h x hx).cexp | lemma | continuous_on.cexp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_on",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.cexp (h : continuous f) : continuous (λ y, exp (f y)) | continuous_iff_continuous_at.2 $ λ x, h.continuous_at.cexp | lemma | continuous.cexp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_exp : continuous exp | complex.continuous_re.comp complex.continuous_of_real.cexp | lemma | real.continuous_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_exp {s : set ℝ} : continuous_on exp s | continuous_exp.continuous_on | lemma | real.continuous_on_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_on",
"continuous_on_exp",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.exp {l : filter α} {f : α → ℝ} {z : ℝ} (hf : tendsto f l (𝓝 z)) :
tendsto (λ x, exp (f x)) l (𝓝 (exp z)) | (continuous_exp.tendsto _).comp hf | lemma | filter.tendsto.exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.exp (h : continuous_within_at f s x) :
continuous_within_at (λ y, exp (f y)) s x | h.exp | lemma | continuous_within_at.exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_within_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.exp (h : continuous_at f x) : continuous_at (λ y, exp (f y)) x | h.exp | lemma | continuous_at.exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.exp (h : continuous_on f s) : continuous_on (λ y, exp (f y)) s | λ x hx, (h x hx).exp | lemma | continuous_on.exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous_on",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.exp (h : continuous f) : continuous (λ y, exp (f y)) | continuous_iff_continuous_at.2 $ λ x, h.continuous_at.exp | lemma | continuous.exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_half (x : ℝ) : exp (x / 2) = sqrt (exp x) | by rw [eq_comm, sqrt_eq_iff_sq_eq, sq, ← exp_add, add_halves]; exact (exp_pos _).le | lemma | real.exp_half | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"add_halves",
"exp",
"exp_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_at_top : tendsto exp at_top at_top | begin
have A : tendsto (λx:ℝ, x + 1) at_top at_top :=
tendsto_at_top_add_const_right at_top 1 tendsto_id,
have B : ∀ᶠ x in at_top, x + 1 ≤ exp x :=
eventually_at_top.2 ⟨0, λx hx, add_one_le_exp x⟩,
exact tendsto_at_top_mono' at_top B A
end | lemma | real.tendsto_exp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | The real exponential function tends to `+∞` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_neg_at_top_nhds_0 : tendsto (λx, exp (-x)) at_top (𝓝 0) | (tendsto_inv_at_top_zero.comp tendsto_exp_at_top).congr (λx, (exp_neg x).symm) | lemma | real.tendsto_exp_neg_at_top_nhds_0 | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"exp_neg"
] | The real exponential function tends to `0` at `-∞` or, equivalently, `exp(-x)` tends to `0`
at `+∞` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_nhds_0_nhds_1 : tendsto exp (𝓝 0) (𝓝 1) | by { convert continuous_exp.tendsto 0, simp } | lemma | real.tendsto_exp_nhds_0_nhds_1 | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | The real exponential function tends to `1` at `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_at_bot : tendsto exp at_bot (𝓝 0) | (tendsto_exp_neg_at_top_nhds_0.comp tendsto_neg_at_bot_at_top).congr $
λ x, congr_arg exp $ neg_neg x | lemma | real.tendsto_exp_at_bot | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_at_bot_nhds_within : tendsto exp at_bot (𝓝[>] 0) | tendsto_inf.2 ⟨tendsto_exp_at_bot, tendsto_principal.2 $ eventually_of_forall exp_pos⟩ | lemma | real.tendsto_exp_at_bot_nhds_within | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_under_ge_exp_comp (l : filter α) (f : α → ℝ) :
is_bounded_under (≥) l (λ x, exp (f x)) | is_bounded_under_of ⟨0, λ x, (exp_pos _).le⟩ | lemma | real.is_bounded_under_ge_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_under_le_exp_comp {f : α → ℝ} :
is_bounded_under (≤) l (λ x, exp (f x)) ↔ is_bounded_under (≤) l f | exp_monotone.is_bounded_under_le_comp tendsto_exp_at_top | lemma | real.is_bounded_under_le_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_div_pow_at_top (n : ℕ) : tendsto (λx, exp x / x^n) at_top at_top | begin
refine (at_top_basis_Ioi.tendsto_iff (at_top_basis' 1)).2 (λ C hC₁, _),
have hC₀ : 0 < C, from zero_lt_one.trans_le hC₁,
have : 0 < (exp 1 * C)⁻¹ := inv_pos.2 (mul_pos (exp_pos _) hC₀),
obtain ⟨N, hN⟩ : ∃ N, ∀ k ≥ N, (↑k ^ n : ℝ) / exp 1 ^ k < (exp 1 * C)⁻¹ :=
eventually_at_top.1 ((tendsto_pow_const_d... | lemma | real.tendsto_exp_div_pow_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"div_eq_inv_mul",
"div_le_div_of_le",
"div_lt_iff",
"exp",
"exp_add",
"gt_mem_nhds",
"le_div_iff",
"le_div_iff'",
"mul_div_mul_left",
"mul_one",
"nat.ceil_lt_add_one",
"nat.le_ceil",
"pow_le_pow_of_le_left",
"pow_pos",
"set.mem_Ici",
"set.mem_Ioi",
"tendsto_pow_const_div_const_pow_of... | The function `exp(x)/x^n` tends to `+∞` at `+∞`, for any natural number `n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_pow_mul_exp_neg_at_top_nhds_0 (n : ℕ) : tendsto (λx, x^n * exp (-x)) at_top (𝓝 0) | (tendsto_inv_at_top_zero.comp (tendsto_exp_div_pow_at_top n)).congr $ λx,
by rw [comp_app, inv_eq_one_div, div_div_eq_mul_div, one_mul, div_eq_mul_inv, exp_neg] | lemma | real.tendsto_pow_mul_exp_neg_at_top_nhds_0 | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"div_div_eq_mul_div",
"div_eq_mul_inv",
"exp",
"exp_neg",
"inv_eq_one_div",
"one_mul"
] | The function `x^n * exp(-x)` tends to `0` at `+∞`, for any natural number `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_mul_exp_add_div_pow_at_top (b c : ℝ) (n : ℕ) (hb : 0 < b) :
tendsto (λ x, (b * exp x + c) / x ^ n) at_top at_top | begin
rcases eq_or_ne n 0 with rfl | hn,
{ simp only [pow_zero, div_one],
exact (tendsto_exp_at_top.const_mul_at_top hb).at_top_add tendsto_const_nhds },
simp only [add_div, mul_div_assoc],
exact ((tendsto_exp_div_pow_at_top n).const_mul_at_top hb).at_top_add
(tendsto_const_nhds.div_at_top (tendsto_pow_... | lemma | real.tendsto_mul_exp_add_div_pow_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"add_div",
"div_one",
"eq_or_ne",
"exp",
"mul_div_assoc",
"pow_zero",
"tendsto_const_nhds"
] | The function `(b * exp x + c) / (x ^ n)` tends to `+∞` at `+∞`, for any natural number
`n` and any real numbers `b` and `c` such that `b` is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_div_pow_mul_exp_add_at_top (b c : ℝ) (n : ℕ) (hb : 0 ≠ b) :
tendsto (λ x, x ^ n / (b * exp x + c)) at_top (𝓝 0) | begin
have H : ∀ d e, 0 < d → tendsto (λ (x:ℝ), x^n / (d * (exp x) + e)) at_top (𝓝 0),
{ intros b' c' h,
convert (tendsto_mul_exp_add_div_pow_at_top b' c' n h).inv_tendsto_at_top ,
ext x,
simpa only [pi.inv_apply] using (inv_div _ _).symm },
cases lt_or_gt_of_ne hb,
{ exact H b c h },
{ convert (... | lemma | real.tendsto_div_pow_mul_exp_add_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"inv_div",
"neg_div_neg_eq",
"pi.inv_apply"
] | The function `(x ^ n) / (b * exp x + c)` tends to `0` at `+∞`, for any natural number
`n` and any real numbers `b` and `c` such that `b` is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_order_iso : ℝ ≃o Ioi (0 : ℝ) | strict_mono.order_iso_of_surjective _ (exp_strict_mono.cod_restrict exp_pos) $
(continuous_exp.subtype_mk _).surjective
(by simp only [tendsto_Ioi_at_top, subtype.coe_mk, tendsto_exp_at_top])
(by simp [tendsto_exp_at_bot_nhds_within]) | def | real.exp_order_iso | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"strict_mono.order_iso_of_surjective",
"subtype.coe_mk"
] | `real.exp` as an order isomorphism between `ℝ` and `(0, +∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_exp_order_iso_apply (x : ℝ) : (exp_order_iso x : ℝ) = exp x | rfl | lemma | real.coe_exp_order_iso_apply | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_exp_order_iso : coe ∘ exp_order_iso = exp | rfl | lemma | real.coe_comp_exp_order_iso | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_exp : range exp = Ioi 0 | by rw [← coe_comp_exp_order_iso, range_comp, exp_order_iso.range_eq, image_univ, subtype.range_coe] | lemma | real.range_exp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp_at_top : map exp at_top = at_top | by rw [← coe_comp_exp_order_iso, ← filter.map_map, order_iso.map_at_top, map_coe_Ioi_at_top] | lemma | real.map_exp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter.map_map",
"order_iso.map_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_exp_at_top : comap exp at_top = at_top | by rw [← map_exp_at_top, comap_map exp_injective, map_exp_at_top] | lemma | real.comap_exp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_comp_at_top {f : α → ℝ} :
tendsto (λ x, exp (f x)) l at_top ↔ tendsto f l at_top | by rw [← tendsto_comap_iff, comap_exp_at_top] | lemma | real.tendsto_exp_comp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_comp_exp_at_top {f : ℝ → α} :
tendsto (λ x, f (exp x)) at_top l ↔ tendsto f at_top l | by rw [← tendsto_map'_iff, map_exp_at_top] | lemma | real.tendsto_comp_exp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp_at_bot : map exp at_bot = 𝓝[>] 0 | by rw [← coe_comp_exp_order_iso, ← filter.map_map, exp_order_iso.map_at_bot, ← map_coe_Ioi_at_bot] | lemma | real.map_exp_at_bot | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter.map_map",
"map_coe_Ioi_at_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_exp_nhds_within_Ioi_zero : comap exp (𝓝[>] 0) = at_bot | by rw [← map_exp_at_bot, comap_map exp_injective] | lemma | real.comap_exp_nhds_within_Ioi_zero | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_comp_exp_at_bot {f : ℝ → α} :
tendsto (λ x, f (exp x)) at_bot l ↔ tendsto f (𝓝[>] 0) l | by rw [← map_exp_at_bot, tendsto_map'_iff] | lemma | real.tendsto_comp_exp_at_bot | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_exp_nhds_zero : comap exp (𝓝 0) = at_bot | (comap_nhds_within_range exp 0).symm.trans $ by simp | lemma | real.comap_exp_nhds_zero | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"comap_nhds_within_range",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_comp_nhds_zero {f : α → ℝ} :
tendsto (λ x, exp (f x)) l (𝓝 0) ↔ tendsto f l at_bot | by rw [← tendsto_comap_iff, comap_exp_nhds_zero] | lemma | real.tendsto_exp_comp_nhds_zero | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_exp_at_top {n : ℕ} : (λ x, x^n) =o[at_top] real.exp | by simpa [is_o_iff_tendsto (λ x hx, ((exp_pos x).ne' hx).elim)]
using tendsto_div_pow_mul_exp_add_at_top 1 0 n zero_ne_one | lemma | real.is_o_pow_exp_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"real.exp",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_exp_comp_exp_comp {f g : α → ℝ} :
(λ x, exp (f x)) =O[l] (λ x, exp (g x)) ↔ is_bounded_under (≤) l (f - g) | iff.trans (is_O_iff_is_bounded_under_le_div $ eventually_of_forall $ λ x, exp_ne_zero _) $
by simp only [norm_eq_abs, abs_exp, ← exp_sub, is_bounded_under_le_exp_comp, pi.sub_def] | lemma | real.is_O_exp_comp_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_exp_comp_exp_comp {f g : α → ℝ} :
(λ x, exp (f x)) =Θ[l] (λ x, exp (g x)) ↔ is_bounded_under (≤) l (λ x, |f x - g x|) | by simp only [is_bounded_under_le_abs, ← is_bounded_under_le_neg, neg_sub, is_Theta,
is_O_exp_comp_exp_comp, pi.sub_def] | lemma | real.is_Theta_exp_comp_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_exp_comp_exp_comp {f g : α → ℝ} :
(λ x, exp (f x)) =o[l] (λ x, exp (g x)) ↔ tendsto (λ x, g x - f x) l at_top | by simp only [is_o_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_at_top_iff, false_implies_iff,
implies_true_iff, tendsto_exp_comp_nhds_zero, neg_sub] | lemma | real.is_o_exp_comp_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_one_exp_comp {f : α → ℝ} :
(λ x, 1 : α → ℝ) =o[l] (λ x, exp (f x)) ↔ tendsto f l at_top | by simp only [← exp_zero, is_o_exp_comp_exp_comp, sub_zero] | lemma | real.is_o_one_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"exp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_one_exp_comp {f : α → ℝ} :
(λ x, 1 : α → ℝ) =O[l] (λ x, exp (f x)) ↔ is_bounded_under (≥) l f | by simp only [← exp_zero, is_O_exp_comp_exp_comp, pi.sub_def, zero_sub, is_bounded_under_le_neg] | lemma | real.is_O_one_exp_comp | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"exp_zero"
] | `real.exp (f x)` is bounded away from zero along a filter if and only if this filter is bounded
from below under `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_exp_comp_one {f : α → ℝ} :
(λ x, exp (f x)) =O[l] (λ x, 1 : α → ℝ) ↔ is_bounded_under (≤) l f | by simp only [is_O_one_iff, norm_eq_abs, abs_exp, is_bounded_under_le_exp_comp] | lemma | real.is_O_exp_comp_one | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | `real.exp (f x)` is bounded away from zero along a filter if and only if this filter is bounded
from below under `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Theta_exp_comp_one {f : α → ℝ} :
(λ x, exp (f x)) =Θ[l] (λ x, 1 : α → ℝ) ↔ is_bounded_under (≤) l (λ x, |f x|) | by simp only [← exp_zero, is_Theta_exp_comp_exp_comp, sub_zero] | lemma | real.is_Theta_exp_comp_one | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"exp_zero"
] | `real.exp (f x)` is bounded away from zero and infinity along a filter `l` if and only if
`|f x|` is bounded from above along this filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_exp_comap_abs_at_top : comap exp (comap abs at_top) = comap re at_top | calc comap exp (comap abs at_top) = comap re (comap real.exp at_top) :
by simp only [comap_comap, (∘), abs_exp]
... = comap re at_top : by rw [real.comap_exp_at_top] | lemma | complex.comap_exp_comap_abs_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"real.comap_exp_at_top",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_exp_nhds_zero : comap exp (𝓝 0) = comap re at_bot | calc comap exp (𝓝 0) = comap re (comap real.exp (𝓝 0)) :
by simp only [comap_comap, ← comap_abs_nhds_zero, (∘), abs_exp]
... = comap re at_bot : by rw [real.comap_exp_nhds_zero] | lemma | complex.comap_exp_nhds_zero | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"real.comap_exp_nhds_zero",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_exp_nhds_within_zero : comap exp (𝓝[≠] 0) = comap re at_bot | have exp ⁻¹' {0}ᶜ = univ, from eq_univ_of_forall exp_ne_zero,
by simp [nhds_within, comap_exp_nhds_zero, this] | lemma | complex.comap_exp_nhds_within_zero | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_nhds_zero_iff {α : Type*} {l : filter α} {f : α → ℂ} :
tendsto (λ x, exp (f x)) l (𝓝 0) ↔ tendsto (λ x, re (f x)) l at_bot | by rw [← tendsto_comap_iff, comap_exp_nhds_zero, tendsto_comap_iff] | lemma | complex.tendsto_exp_nhds_zero_iff | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_exp_comap_re_at_top : tendsto exp (comap re at_top) (comap abs at_top) | comap_exp_comap_abs_at_top ▸ tendsto_comap | lemma | complex.tendsto_exp_comap_re_at_top | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | `complex.abs (complex.exp z) → ∞` as `complex.re z → ∞`. TODO: use `bornology.cobounded`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_comap_re_at_bot : tendsto exp (comap re at_bot) (𝓝 0) | comap_exp_nhds_zero ▸ tendsto_comap | lemma | complex.tendsto_exp_comap_re_at_bot | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | `complex.exp z → 0` as `complex.re z → -∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_exp_comap_re_at_bot_nhds_within : tendsto exp (comap re at_bot) (𝓝[≠] 0) | comap_exp_nhds_within_zero ▸ tendsto_comap | lemma | complex.tendsto_exp_comap_re_at_bot_nhds_within | analysis.special_functions | src/analysis/special_functions/exp.lean | [
"analysis.asymptotics.theta",
"analysis.complex.basic",
"analysis.specific_limits.normed"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_zero_of_radius_pos (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 | begin
convert (has_fpower_series_at_exp_zero_of_radius_pos h).has_strict_fderiv_at,
ext x,
change x = exp_series 𝕂 𝔸 1 (λ _, x),
simp [exp_series_apply_eq]
end | lemma | has_strict_fderiv_at_exp_zero_of_radius_pos | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"exp_series",
"exp_series_apply_eq",
"has_fpower_series_at_exp_zero_of_radius_pos",
"has_strict_fderiv_at"
] | The exponential in a Banach-algebra `𝔸` over a normed field `𝕂` has strict Fréchet-derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero, as long as it converges on a neighborhood of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_exp_zero_of_radius_pos (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 | (has_strict_fderiv_at_exp_zero_of_radius_pos h).has_fderiv_at | lemma | has_fderiv_at_exp_zero_of_radius_pos | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"exp_series",
"has_fderiv_at",
"has_strict_fderiv_at_exp_zero_of_radius_pos"
] | The exponential in a Banach-algebra `𝔸` over a normed field `𝕂` has Fréchet-derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero, as long as it converges on a neighborhood of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_exp_of_mem_ball [char_zero 𝕂] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (exp 𝕂) (exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸) x | begin
have hpos : 0 < (exp_series 𝕂 𝔸).radius := (zero_le _).trans_lt hx,
rw has_fderiv_at_iff_is_o_nhds_zero,
suffices : (λ h, exp 𝕂 x * (exp 𝕂 (0 + h) - exp 𝕂 0 - continuous_linear_map.id 𝕂 𝔸 h))
=ᶠ[𝓝 0] (λ h, exp 𝕂 (x + h) - exp 𝕂 x - exp 𝕂 x • continuous_linear_map.id 𝕂 𝔸 h),
{ refine (is_o... | lemma | has_fderiv_at_exp_of_mem_ball | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"char_zero",
"continuous_linear_map.id",
"continuous_linear_map.id_apply",
"emetric.ball",
"emetric.ball_mem_nhds",
"exp",
"exp_add_of_mem_ball",
"exp_series",
"exp_zero",
"has_fderiv_at",
"has_fderiv_at_exp_zero_of_radius_pos",
"has_fderiv_at_iff_is_o_nhds_zero",
"ring",
"smul_eq_mul"
] | The exponential map in a commutative Banach-algebra `𝔸` over a normed field `𝕂` of
characteristic zero has Fréchet-derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x` in the
disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_exp_of_mem_ball [char_zero 𝕂] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂) (exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸) x | let ⟨p, hp⟩ := analytic_at_exp_of_mem_ball x hx in
hp.has_fderiv_at.unique (has_fderiv_at_exp_of_mem_ball hx) ▸ hp.has_strict_fderiv_at | lemma | has_strict_fderiv_at_exp_of_mem_ball | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"analytic_at_exp_of_mem_ball",
"char_zero",
"emetric.ball",
"exp",
"exp_series",
"has_fderiv_at_exp_of_mem_ball",
"has_strict_fderiv_at"
] | The exponential map in a commutative Banach-algebra `𝔸` over a normed field `𝕂` of
characteristic zero has strict Fréchet-derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x` in
the disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_exp_of_mem_ball [char_zero 𝕂] {x : 𝕂}
(hx : x ∈ emetric.ball (0 : 𝕂) (exp_series 𝕂 𝕂).radius) :
has_strict_deriv_at (exp 𝕂) (exp 𝕂 x) x | by simpa using (has_strict_fderiv_at_exp_of_mem_ball hx).has_strict_deriv_at | lemma | has_strict_deriv_at_exp_of_mem_ball | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"char_zero",
"emetric.ball",
"exp",
"exp_series",
"has_strict_deriv_at",
"has_strict_fderiv_at_exp_of_mem_ball"
] | The exponential map in a complete normed field `𝕂` of characteristic zero has strict derivative
`exp 𝕂 x` at any point `x` in the disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_exp_of_mem_ball [char_zero 𝕂] {x : 𝕂}
(hx : x ∈ emetric.ball (0 : 𝕂) (exp_series 𝕂 𝕂).radius) :
has_deriv_at (exp 𝕂) (exp 𝕂 x) x | (has_strict_deriv_at_exp_of_mem_ball hx).has_deriv_at | lemma | has_deriv_at_exp_of_mem_ball | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"char_zero",
"emetric.ball",
"exp",
"exp_series",
"has_deriv_at",
"has_strict_deriv_at_exp_of_mem_ball"
] | The exponential map in a complete normed field `𝕂` of characteristic zero has derivative
`exp 𝕂 x` at any point `x` in the disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_exp_zero_of_radius_pos (h : 0 < (exp_series 𝕂 𝕂).radius) :
has_strict_deriv_at (exp 𝕂) (1 : 𝕂) 0 | (has_strict_fderiv_at_exp_zero_of_radius_pos h).has_strict_deriv_at | lemma | has_strict_deriv_at_exp_zero_of_radius_pos | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"exp_series",
"has_strict_deriv_at",
"has_strict_fderiv_at_exp_zero_of_radius_pos"
] | The exponential map in a complete normed field `𝕂` of characteristic zero has strict derivative
`1` at zero, as long as it converges on a neighborhood of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_exp_zero_of_radius_pos (h : 0 < (exp_series 𝕂 𝕂).radius) :
has_deriv_at (exp 𝕂) (1 : 𝕂) 0 | (has_strict_deriv_at_exp_zero_of_radius_pos h).has_deriv_at | lemma | has_deriv_at_exp_zero_of_radius_pos | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"exp_series",
"has_deriv_at",
"has_strict_deriv_at_exp_zero_of_radius_pos"
] | The exponential map in a complete normed field `𝕂` of characteristic zero has derivative
`1` at zero, as long as it converges on a neighborhood of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_exp_zero :
has_strict_fderiv_at (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 | has_strict_fderiv_at_exp_zero_of_radius_pos (exp_series_radius_pos 𝕂 𝔸) | lemma | has_strict_fderiv_at_exp_zero | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"exp_series_radius_pos",
"has_strict_fderiv_at",
"has_strict_fderiv_at_exp_zero_of_radius_pos"
] | The exponential in a Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has strict Fréchet-derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_exp_zero :
has_fderiv_at (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 | has_strict_fderiv_at_exp_zero.has_fderiv_at | lemma | has_fderiv_at_exp_zero | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"has_fderiv_at"
] | The exponential in a Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has Fréchet-derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_exp {x : 𝔸} :
has_strict_fderiv_at (exp 𝕂) (exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸) x | has_strict_fderiv_at_exp_of_mem_ball ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | has_strict_fderiv_at_exp | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"edist_lt_top",
"exp",
"exp_series_radius_eq_top",
"has_strict_fderiv_at",
"has_strict_fderiv_at_exp_of_mem_ball"
] | The exponential map in a commutative Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has strict
Fréchet-derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_exp {x : 𝔸} :
has_fderiv_at (exp 𝕂) (exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸) x | has_strict_fderiv_at_exp.has_fderiv_at | lemma | has_fderiv_at_exp | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"exp",
"has_fderiv_at"
] | The exponential map in a commutative Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has
Fréchet-derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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