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_strict_deriv_at_exp {x : 𝕂} : has_strict_deriv_at (exp 𝕂) (exp 𝕂 x) x | has_strict_deriv_at_exp_of_mem_ball ((exp_series_radius_eq_top 𝕂 𝕂).symm ▸ edist_lt_top _ _) | lemma | has_strict_deriv_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_deriv_at",
"has_strict_deriv_at_exp_of_mem_ball"
] | The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has strict derivative `exp 𝕂 x` at any point
`x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_exp {x : 𝕂} : has_deriv_at (exp 𝕂) (exp 𝕂 x) x | has_strict_deriv_at_exp.has_deriv_at | lemma | has_deriv_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_deriv_at"
] | The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has derivative `exp 𝕂 x` at any point `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_exp_zero : has_strict_deriv_at (exp 𝕂) (1 : 𝕂) 0 | has_strict_deriv_at_exp_zero_of_radius_pos (exp_series_radius_pos 𝕂 𝕂) | lemma | has_strict_deriv_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_deriv_at",
"has_strict_deriv_at_exp_zero_of_radius_pos"
] | The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has strict derivative `1` at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_exp_zero :
has_deriv_at (exp 𝕂) (1 : 𝕂) 0 | has_strict_deriv_at_exp_zero.has_deriv_at | lemma | has_deriv_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_deriv_at"
] | The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has derivative `1` at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.exp_eq_exp_ℂ : complex.exp = exp ℂ | begin
refine funext (λ x, _),
rw [complex.exp, exp_eq_tsum_div],
exact tendsto_nhds_unique x.exp'.tendsto_limit
(exp_series_div_summable ℝ x).has_sum.tendsto_sum_nat
end | lemma | complex.exp_eq_exp_ℂ | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"complex.exp",
"exp",
"exp_eq_tsum_div",
"exp_series_div_summable",
"has_sum.tendsto_sum_nat",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.exp_eq_exp_ℝ : real.exp = exp ℝ | by { ext x, exact_mod_cast congr_fun complex.exp_eq_exp_ℂ x } | lemma | real.exp_eq_exp_ℝ | analysis.special_functions | src/analysis/special_functions/exponential.lean | [
"analysis.normed_space.exponential",
"analysis.calculus.fderiv_analytic",
"topology.metric_space.cau_seq_filter"
] | [
"complex.exp_eq_exp_ℂ",
"exp",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_exp_smul_const_of_mem_ball
(x : 𝔸) (t : 𝕊) (htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smul_right x) t | begin
-- TODO: prove this via `has_fderiv_at_exp_of_mem_ball` using the commutative ring
-- `algebra.elemental_algebra 𝕊 x`. See leanprover-community/mathlib#19062 for discussion.
have hpos : 0 < (exp_series 𝕂 𝔸).radius := (zero_le _).trans_lt htx,
rw has_fderiv_at_iff_is_o_nhds_zero,
suffices :
(λ h, ... | lemma | has_fderiv_at_exp_smul_const_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"
] | [
"add_smul",
"commute",
"commute.refl",
"continuous_linear_map.one_apply",
"continuous_linear_map.smul_apply",
"continuous_linear_map.smul_right_apply",
"emetric.ball",
"emetric.ball_mem_nhds",
"exp",
"exp_add_of_commute_of_mem_ball",
"exp_series",
"exp_zero",
"has_fderiv_at",
"has_fderiv_a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_exp_smul_const_of_mem_ball'
(x : 𝔸) (t : 𝕊) (htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).smul_right x).smul_right (exp 𝕂 (t • x))) t | begin
convert has_fderiv_at_exp_smul_const_of_mem_ball 𝕂 _ _ htx using 1,
ext t',
show commute (t' • x) (exp 𝕂 (t • x)),
exact (((commute.refl x).smul_left t').smul_right t).exp_right 𝕂,
end | lemma | has_fderiv_at_exp_smul_const_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"
] | [
"commute",
"commute.refl",
"emetric.ball",
"exp",
"exp_series",
"has_fderiv_at",
"has_fderiv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕊)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smul_right x) t | let ⟨p, hp⟩ := analytic_at_exp_of_mem_ball (t • x) htx in
have deriv₁ : has_strict_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x)) _ t,
from hp.has_strict_fderiv_at.comp t
((continuous_linear_map.id 𝕂 𝕊).smul_right x).has_strict_fderiv_at,
have deriv₂ : has_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x)) _ t,
from has_fderiv_at_ex... | lemma | has_strict_fderiv_at_exp_smul_const_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",
"continuous_linear_map.id",
"emetric.ball",
"exp",
"exp_series",
"has_fderiv_at",
"has_fderiv_at_exp_smul_const_of_mem_ball",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).smul_right x).smul_right (exp 𝕂 (t • x))) t | let ⟨p, hp⟩ := analytic_at_exp_of_mem_ball (t • x) htx in
begin
convert has_strict_fderiv_at_exp_smul_const_of_mem_ball 𝕂 _ _ htx using 1,
ext t',
show commute (t' • x) (exp 𝕂 (t • x)),
exact (((commute.refl x).smul_left t').smul_right t).exp_right 𝕂,
end | lemma | has_strict_fderiv_at_exp_smul_const_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",
"commute",
"commute.refl",
"emetric.ball",
"exp",
"exp_series",
"has_strict_fderiv_at",
"has_strict_fderiv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_strict_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t | by simpa using (has_strict_fderiv_at_exp_smul_const_of_mem_ball 𝕂 x t htx).has_strict_deriv_at | lemma | has_strict_deriv_at_exp_smul_const_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"
] | [
"emetric.ball",
"exp",
"exp_series",
"has_strict_deriv_at",
"has_strict_fderiv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_strict_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t | by simpa using (has_strict_fderiv_at_exp_smul_const_of_mem_ball' 𝕂 x t htx).has_strict_deriv_at | lemma | has_strict_deriv_at_exp_smul_const_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"
] | [
"emetric.ball",
"exp",
"exp_series",
"has_strict_deriv_at",
"has_strict_fderiv_at_exp_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t | (has_strict_deriv_at_exp_smul_const_of_mem_ball x t htx).has_deriv_at | lemma | has_deriv_at_exp_smul_const_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"
] | [
"emetric.ball",
"exp",
"exp_series",
"has_deriv_at",
"has_strict_deriv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
(htx : t • x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t | (has_strict_deriv_at_exp_smul_const_of_mem_ball' x t htx).has_deriv_at | lemma | has_deriv_at_exp_smul_const_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"
] | [
"emetric.ball",
"exp",
"exp_series",
"has_deriv_at",
"has_strict_deriv_at_exp_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_exp_smul_const (x : 𝔸) (t : 𝕊) :
has_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smul_right x) t | has_fderiv_at_exp_smul_const_of_mem_ball 𝕂 _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_fderiv_at_exp_smul_const | 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_fderiv_at",
"has_fderiv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_exp_smul_const' (x : 𝔸) (t : 𝕊) :
has_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).smul_right x).smul_right (exp 𝕂 (t • x))) t | has_fderiv_at_exp_smul_const_of_mem_ball' 𝕂 _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_fderiv_at_exp_smul_const' | 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_fderiv_at",
"has_fderiv_at_exp_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_smul_const (x : 𝔸) (t : 𝕊) :
has_strict_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smul_right x) t | has_strict_fderiv_at_exp_smul_const_of_mem_ball 𝕂 _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_strict_fderiv_at_exp_smul_const | 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_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_smul_const' (x : 𝔸) (t : 𝕊) :
has_strict_fderiv_at (λ u : 𝕊, exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).smul_right x).smul_right (exp 𝕂 (t • x))) t | has_strict_fderiv_at_exp_smul_const_of_mem_ball' 𝕂 _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_strict_fderiv_at_exp_smul_const' | 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_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp_smul_const (x : 𝔸) (t : 𝕂) :
has_strict_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t | has_strict_deriv_at_exp_smul_const_of_mem_ball _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_strict_deriv_at_exp_smul_const | 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_deriv_at",
"has_strict_deriv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp_smul_const' (x : 𝔸) (t : 𝕂) :
has_strict_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t | has_strict_deriv_at_exp_smul_const_of_mem_ball' _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_strict_deriv_at_exp_smul_const' | 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_deriv_at",
"has_strict_deriv_at_exp_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp_smul_const (x : 𝔸) (t : 𝕂) :
has_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t | has_deriv_at_exp_smul_const_of_mem_ball _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_deriv_at_exp_smul_const | 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_deriv_at",
"has_deriv_at_exp_smul_const_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp_smul_const' (x : 𝔸) (t : 𝕂) :
has_deriv_at (λ u : 𝕂, exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t | has_deriv_at_exp_smul_const_of_mem_ball' _ _ $
(exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | has_deriv_at_exp_smul_const' | 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_deriv_at",
"has_deriv_at_exp_smul_const_of_mem_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp (x : ℂ) : has_deriv_at exp (exp x) x | begin
rw has_deriv_at_iff_is_o_nhds_zero,
have : (1 : ℕ) < 2 := by norm_num,
refine (is_O.of_bound (‖exp x‖) _).trans_is_o (is_o_pow_id this),
filter_upwards [metric.ball_mem_nhds (0 : ℂ) zero_lt_one],
simp only [metric.mem_ball, dist_zero_right, norm_pow],
exact λ z hz, exp_bound_sq x z hz.le,
end | lemma | complex.has_deriv_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"exp",
"has_deriv_at",
"has_deriv_at_exp",
"has_deriv_at_iff_is_o_nhds_zero",
"metric.ball_mem_nhds",
"metric.mem_ball",
"norm_pow",
"zero_lt_one"
] | The complex exponential is everywhere differentiable, with the derivative `exp x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_exp : differentiable 𝕜 exp | λ x, (has_deriv_at_exp x).differentiable_at.restrict_scalars 𝕜 | lemma | complex.differentiable_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable",
"differentiable_at.restrict_scalars",
"exp",
"has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_exp {x : ℂ} : differentiable_at 𝕜 exp x | differentiable_exp x | lemma | complex.differentiable_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_exp : deriv exp = exp | funext $ λ x, (has_deriv_at_exp x).deriv | lemma | complex.deriv_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"deriv",
"deriv_exp",
"exp",
"has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_exp : ∀ n : ℕ, (deriv^[n] exp) = exp | | 0 := rfl
| (n+1) := by rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n] | lemma | complex.iter_deriv_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"deriv",
"deriv_exp",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_exp : ∀ {n}, cont_diff 𝕜 n exp | begin
refine cont_diff_all_iff_nat.2 (λ n, _),
have : cont_diff ℂ ↑n exp,
{ induction n with n ihn,
{ exact cont_diff_zero.2 continuous_exp },
{ rw cont_diff_succ_iff_deriv,
use differentiable_exp,
rwa deriv_exp }, },
exact this.restrict_scalars 𝕜
end | lemma | complex.cont_diff_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff",
"cont_diff_succ_iff_deriv",
"deriv_exp",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp (x : ℂ) : has_strict_deriv_at exp (exp x) x | cont_diff_exp.cont_diff_at.has_strict_deriv_at' (has_deriv_at_exp x) le_rfl | lemma | complex.has_strict_deriv_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"exp",
"has_deriv_at_exp",
"has_strict_deriv_at",
"has_strict_deriv_at_exp",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_exp_real (x : ℂ) :
has_strict_fderiv_at exp (exp x • (1 : ℂ →L[ℝ] ℂ)) x | (has_strict_deriv_at_exp x).complex_to_real_fderiv | lemma | complex.has_strict_fderiv_at_exp_real | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"exp",
"has_strict_deriv_at_exp",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.cexp (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') x | (complex.has_strict_deriv_at_exp (f x)).comp x hf | lemma | has_strict_deriv_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"complex.has_strict_deriv_at_exp",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.cexp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') x | (complex.has_deriv_at_exp (f x)).comp x hf | lemma | has_deriv_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"complex.has_deriv_at_exp",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.cexp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') s x | (complex.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"complex.has_deriv_at_exp",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_cexp (hf : differentiable_within_at 𝕜 f s x)
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λ x, complex.exp (f x)) s x = complex.exp (f x) * deriv_within f s x | hf.has_deriv_within_at.cexp.deriv_within hxs | lemma | deriv_within_cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_cexp (hc : differentiable_at 𝕜 f x) :
deriv (λ x, complex.exp (f x)) x = complex.exp (f x) * deriv f x | hc.has_deriv_at.cexp.deriv | lemma | deriv_cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.cexp (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') x | (complex.has_strict_deriv_at_exp (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"complex.has_strict_deriv_at_exp",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.cexp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') s x | (complex.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"complex.has_deriv_at_exp",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.cexp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') x | has_fderiv_within_at_univ.1 $ hf.has_fderiv_within_at.cexp | lemma | has_fderiv_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.cexp (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λ x, complex.exp (f x)) s x | hf.has_fderiv_within_at.cexp.differentiable_within_at | lemma | differentiable_within_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.cexp (hc : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λ x, complex.exp (f x)) x | hc.has_fderiv_at.cexp.differentiable_at | lemma | differentiable_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.cexp (hc : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λ x, complex.exp (f x)) s | λ x h, (hc x h).cexp | lemma | differentiable_on.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.cexp (hc : differentiable 𝕜 f) :
differentiable 𝕜 (λ x, complex.exp (f x)) | λ x, (hc x).cexp | lemma | differentiable.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cexp {n} (h : cont_diff 𝕜 n f) :
cont_diff 𝕜 n (λ x, complex.exp (f x)) | complex.cont_diff_exp.comp h | lemma | cont_diff.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.cexp {n} (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ x, complex.exp (f x)) x | complex.cont_diff_exp.cont_diff_at.comp x hf | lemma | cont_diff_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.cexp {n} (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ x, complex.exp (f x)) s | complex.cont_diff_exp.comp_cont_diff_on hf | lemma | cont_diff_on.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cexp {n} (hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (λ x, complex.exp (f x)) s x | complex.cont_diff_exp.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.cexp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.exp",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_exp (x : ℝ) : has_strict_deriv_at exp (exp x) x | (complex.has_strict_deriv_at_exp x).real_of_complex | lemma | real.has_strict_deriv_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.has_strict_deriv_at_exp",
"exp",
"has_strict_deriv_at",
"has_strict_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_exp (x : ℝ) : has_deriv_at exp (exp x) x | (complex.has_deriv_at_exp x).real_of_complex | lemma | real.has_deriv_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"complex.has_deriv_at_exp",
"exp",
"has_deriv_at",
"has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_exp {n} : cont_diff ℝ n exp | complex.cont_diff_exp.real_of_complex | lemma | real.cont_diff_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_exp : differentiable ℝ exp | λx, (has_deriv_at_exp x).differentiable_at | lemma | real.differentiable_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable",
"differentiable_at",
"exp",
"has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_exp : differentiable_at ℝ exp x | differentiable_exp x | lemma | real.differentiable_at_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_at",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.exp (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.exp (f x)) (real.exp (f x) * f') x | (real.has_strict_deriv_at_exp (f x)).comp x hf | lemma | has_strict_deriv_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_strict_deriv_at",
"real.exp",
"real.has_strict_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.exp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.exp (f x)) (real.exp (f x) * f') x | (real.has_deriv_at_exp (f x)).comp x hf | lemma | has_deriv_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_deriv_at",
"real.exp",
"real.has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.exp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.exp (f x)) (real.exp (f x) * f') s x | (real.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_deriv_within_at",
"real.exp",
"real.has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.exp (f x)) s x = real.exp (f x) * (deriv_within f s x) | hf.has_deriv_within_at.exp.deriv_within hxs | lemma | deriv_within_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"deriv_within",
"differentiable_within_at",
"real.exp",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_exp (hc : differentiable_at ℝ f x) :
deriv (λx, real.exp (f x)) x = real.exp (f x) * (deriv f x) | hc.has_deriv_at.exp.deriv | lemma | deriv_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"deriv",
"differentiable_at",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.exp {n} (hf : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.exp (f x)) | real.cont_diff_exp.comp hf | lemma | cont_diff.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.exp {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.exp (f x)) x | real.cont_diff_exp.cont_diff_at.comp x hf | lemma | cont_diff_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff_at",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.exp {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.exp (f x)) s | real.cont_diff_exp.comp_cont_diff_on hf | lemma | cont_diff_on.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff_on",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.exp {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.exp (f x)) s x | real.cont_diff_exp.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"cont_diff_within_at",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.exp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.exp (f x)) (real.exp (f x) • f') s x | (real.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_fderiv_within_at",
"real.exp",
"real.has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.exp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.exp (f x)) (real.exp (f x) • f') x | (real.has_deriv_at_exp (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_fderiv_at",
"real.exp",
"real.has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.exp (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.exp (f x)) (real.exp (f x) • f') x | (real.has_strict_deriv_at_exp (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"has_strict_fderiv_at",
"real.exp",
"real.has_strict_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.exp (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.exp (f x)) s x | hf.has_fderiv_within_at.exp.differentiable_within_at | lemma | differentiable_within_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_within_at",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.exp (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.exp (f x)) x | hc.has_fderiv_at.exp.differentiable_at | lemma | differentiable_at.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_at",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.exp (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.exp (f x)) s | λ x h, (hc x h).exp | lemma | differentiable_on.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_on",
"exp",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.exp (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.exp (f x)) | λ x, (hc x).exp | lemma | differentiable.exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable",
"exp",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.exp (f x)) s x = real.exp (f x) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.exp.fderiv_within hxs | lemma | fderiv_within_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_within_at",
"fderiv_within",
"real.exp",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_exp (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.exp (f x)) x = real.exp (f x) • (fderiv ℝ f x) | hc.has_fderiv_at.exp.fderiv | lemma | fderiv_exp | analysis.special_functions | src/analysis/special_functions/exp_deriv.lean | [
"analysis.complex.real_deriv"
] | [
"differentiable_at",
"fderiv",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_neg_mul_sq_is_o_exp_neg {b : ℝ} (hb : 0 < b) :
(λ x:ℝ, exp (-b * x^2)) =o[at_top] (λ x:ℝ, exp (-x)) | begin
have A : (λ (x : ℝ), -x - -b * x ^ 2) = (λ x, x * (b * x + (- 1))), by { ext x, ring },
rw [is_o_exp_comp_exp_comp, A],
apply tendsto.at_top_mul_at_top tendsto_id,
apply tendsto_at_top_add_const_right at_top (-1 : ℝ),
exact tendsto.const_mul_at_top hb tendsto_id,
end | lemma | exp_neg_mul_sq_is_o_exp_neg | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_mul_exp_neg_mul_sq_is_o_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
(λ x:ℝ, x ^ s * exp (-b * x^2)) =o[at_top] (λ x:ℝ, exp (-(1/2) * x)) | begin
apply ((is_O_refl (λ x:ℝ, x ^ s) at_top).mul_is_o (exp_neg_mul_sq_is_o_exp_neg hb)).trans,
convert Gamma_integrand_is_o s,
simp_rw [mul_comm],
end | lemma | rpow_mul_exp_neg_mul_sq_is_o_exp_neg | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp",
"exp_neg_mul_sq_is_o_exp_neg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_on_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
integrable_on (λ x:ℝ, x ^ s * exp (-b * x^2)) (Ioi 0) | begin
rw [← Ioc_union_Ioi_eq_Ioi (zero_le_one : (0 : ℝ) ≤ 1), integrable_on_union],
split,
{ rw [←integrable_on_Icc_iff_integrable_on_Ioc],
refine integrable_on.mul_continuous_on _ _ is_compact_Icc,
{ refine (interval_integrable_iff_integrable_Icc_of_le zero_le_one).mp _,
exact interval_integral.int... | lemma | integrable_on_rpow_mul_exp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"continuous_at",
"continuous_on",
"continuous_pow",
"continuous_within_at",
"exp",
"integrable_of_is_O_exp_neg",
"interval_integrable_iff_integrable_Icc_of_le",
"interval_integral.interval_integrable_rpow'",
"rpow_mul_exp_neg_mul_sq_is_o_exp_neg",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
integrable (λ x:ℝ, x ^ s * exp (-b * x^2)) | begin
rw [← integrable_on_univ, ← @Iio_union_Ici _ _ (0 : ℝ), integrable_on_union,
integrable_on_Ici_iff_integrable_on_Ioi],
refine ⟨_, integrable_on_rpow_mul_exp_neg_mul_sq hb hs⟩,
rw ← (measure.measure_preserving_neg (volume : measure ℝ)).integrable_on_comp_preimage
((homeomorph.neg ℝ).to_measurable_e... | lemma | integrable_rpow_mul_exp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"abs_mul",
"abs_of_nonneg",
"exp",
"integrable_on_Ici_iff_integrable_on_Ioi",
"integrable_on_rpow_mul_exp_neg_mul_sq",
"measurable.ae_strongly_measurable",
"measurable_const",
"measurable_set",
"measurable_set_Ioi",
"mul_le_mul_of_nonneg_right",
"neg_sq",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
integrable (λ x:ℝ, exp (-b * x^2)) | by simpa using integrable_rpow_mul_exp_neg_mul_sq hb (by norm_num : (-1 : ℝ) < 0) | lemma | integrable_exp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp",
"integrable_rpow_mul_exp_neg_mul_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_on_Ioi_exp_neg_mul_sq_iff {b : ℝ} :
integrable_on (λ x:ℝ, exp (-b * x^2)) (Ioi 0) ↔ 0 < b | begin
refine ⟨λ h, _, λ h, (integrable_exp_neg_mul_sq h).integrable_on⟩,
by_contra' hb,
have : ∫⁻ x:ℝ in Ioi 0, 1 ≤ ∫⁻ x:ℝ in Ioi 0, ‖exp (-b * x^2)‖₊,
{ apply lintegral_mono (λ x, _),
simp only [neg_mul, ennreal.one_le_coe_iff, ← to_nnreal_one, to_nnreal_le_iff_le_coe,
real.norm_of_nonneg (exp_pos _)... | lemma | integrable_on_Ioi_exp_neg_mul_sq_iff | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"ennreal.one_le_coe_iff",
"exp",
"integrable_exp_neg_mul_sq",
"mul_nonpos_of_nonpos_of_nonneg",
"neg_mul",
"real.norm_of_nonneg",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_exp_neg_mul_sq_iff {b : ℝ} : integrable (λ x:ℝ, exp (-b * x^2)) ↔ 0 < b | ⟨λ h, integrable_on_Ioi_exp_neg_mul_sq_iff.mp h.integrable_on, integrable_exp_neg_mul_sq⟩ | lemma | integrable_exp_neg_mul_sq_iff | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) : integrable (λ x:ℝ, x * exp (-b * x^2)) | by simpa using integrable_rpow_mul_exp_neg_mul_sq hb (by norm_num : (-1 : ℝ) < 1) | lemma | integrable_mul_exp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp",
"integrable_rpow_mul_exp_neg_mul_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cexp_neg_mul_sq (b : ℂ) (x : ℝ) : ‖complex.exp (-b * x^2)‖ = exp (-b.re * x^2) | by rw [complex.norm_eq_abs, complex.abs_exp, ←of_real_pow, mul_comm (-b) _, of_real_mul_re,
neg_re, mul_comm] | lemma | norm_cexp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.abs_exp",
"complex.norm_eq_abs",
"exp",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) : integrable (λ x:ℝ, cexp (-b * x^2)) | begin
refine ⟨(complex.continuous_exp.comp
(continuous_const.mul (continuous_of_real.pow 2))).ae_strongly_measurable, _⟩,
rw ←has_finite_integral_norm_iff,
simp_rw norm_cexp_neg_mul_sq,
exact (integrable_exp_neg_mul_sq hb).2,
end | lemma | integrable_cexp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"integrable_exp_neg_mul_sq",
"norm_cexp_neg_mul_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
integrable (λ x:ℝ, ↑x * cexp (-b * x^2)) | begin
refine ⟨(continuous_of_real.mul (complex.continuous_exp.comp _)).ae_strongly_measurable, _⟩,
{ exact continuous_const.mul (continuous_of_real.pow 2) },
have := (integrable_mul_exp_neg_mul_sq hb).has_finite_integral,
rw ←has_finite_integral_norm_iff at this ⊢,
convert this,
ext1 x,
rw [norm_mul, norm... | lemma | integrable_mul_cexp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.norm_eq_abs",
"integrable_mul_exp_neg_mul_sq",
"norm_cexp_neg_mul_sq",
"norm_mul",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
∫ r:ℝ in Ioi 0, (r : ℂ) * cexp (-b * r ^ 2) = (2 * b)⁻¹ | begin
have hb' : b ≠ 0 := by { contrapose! hb, rw [hb, zero_re], },
have A : ∀ x:ℂ, has_deriv_at (λ x, - (2 * b)⁻¹ * cexp (-b * x^2)) (x * cexp (- b * x^2)) x,
{ intro x,
convert (((has_deriv_at_pow 2 x)).const_mul (-b)).cexp.const_mul (- (2 * b)⁻¹) using 1,
field_simp [hb'],
ring },
have B : tendst... | lemma | integral_mul_cexp_neg_mul_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"bit0_eq_zero",
"complex.exp_zero",
"has_deriv_at",
"has_deriv_at_pow",
"integrable_mul_cexp_neg_mul_sq",
"mul_one",
"mul_zero",
"norm_cexp_neg_mul_sq",
"ring",
"two_ne_zero",
"zero_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_gaussian_sq_complex {b : ℂ} (hb : 0 < b.re) :
(∫ x:ℝ, cexp (-b * x^2)) ^ 2 = π / b | begin
/- We compute `(∫ exp (-b x^2))^2` as an integral over `ℝ^2`, and then make a polar change
of coordinates. We are left with `∫ r * exp (-b r^2)`, which has been computed in
`integral_mul_cexp_neg_mul_sq` using the fact that this function has an obvious primitive. -/
calc
(∫ x:ℝ, cexp (-b * (x:ℂ)^2)) ^ 2... | lemma | integral_gaussian_sq_complex | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.exp_add",
"ennreal.to_real_of_real",
"integral_comp_polar_coord_symm",
"integral_mul_cexp_neg_mul_sq",
"measurable_set_Ioo",
"mul_one",
"one_mul",
"pow_two",
"real.pi_pos",
"ring"
] | The *square* of the Gaussian integral `∫ x:ℝ, exp (-b * x^2)` is equal to `π / b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_gaussian (b : ℝ) : ∫ x, exp (-b * x^2) = sqrt (π / b) | begin
/- First we deal with the crazy case where `b ≤ 0`: then both sides vanish. -/
rcases le_or_lt b 0 with hb|hb,
{ rw [integral_undef, sqrt_eq_zero_of_nonpos],
{ exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb },
{ simpa only [not_lt, integrable_exp_neg_mul_sq_iff] using hb } },
/- Assume now `b >... | theorem | integral_gaussian | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"div_nonpos_of_nonneg_of_nonpos",
"div_pos",
"exp",
"integrable_exp_neg_mul_sq_iff",
"integral_gaussian_sq_complex",
"sq_eq_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_gaussian_integral (b : ℂ) (hb : 0 < re b) :
continuous_at (λ c:ℂ, ∫ x:ℝ, cexp (-c * x^2)) b | begin
let f : ℂ → ℝ → ℂ := λ (c : ℂ) (x : ℝ), cexp (-c * x ^ 2),
obtain ⟨d, hd, hd'⟩ := exists_between hb,
have f_meas : ∀ (c:ℂ), ae_strongly_measurable (f c) volume := λ c, by
{ apply continuous.ae_strongly_measurable,
exact complex.continuous_exp.comp (continuous_const.mul (continuous_of_real.pow 2)) },
... | lemma | continuous_at_gaussian_integral | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"continuous.ae_strongly_measurable",
"continuous_at",
"continuous_const",
"exists_between",
"exp",
"integrable_exp_neg_mul_sq",
"is_open_Ioi",
"mul_le_mul_of_nonneg_right",
"norm_cexp_neg_mul_sq",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_gaussian_complex {b : ℂ} (hb : 0 < re b) :
∫ x:ℝ, cexp (-b * x^2) = (π / b) ^ (1 / 2 : ℂ) | begin
have nv : ∀ {b : ℂ}, (0 < re b) → (b ≠ 0),
{ intros b hb, contrapose! hb, rw hb, simp },
refine (convex_halfspace_re_gt 0).is_preconnected.eq_of_sq_eq
_ _ (λ c hc, _) (λ c hc, _) (by simp : 0 < re (1 : ℂ)) _ hb,
{ -- integral is continuous
exact continuous_at.continuous_on continuous_at_gaussian_i... | theorem | integral_gaussian_complex | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"continuous_at.continuous_on",
"continuous_at_cpow_const",
"continuous_at_gaussian_integral",
"continuous_at_id",
"convex_halfspace_re_gt",
"div_ne_zero",
"div_one",
"div_pos",
"exp",
"integral_gaussian",
"integral_gaussian_sq_complex",
"integral_of_real",
"is_preconnected.eq_of_sq_eq",
"n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_gaussian_complex_Ioi {b : ℂ} (hb : 0 < re b) :
∫ x:ℝ in Ioi 0, cexp (-b * x^2) = (π / b) ^ (1 / 2 : ℂ) / 2 | begin
have full_integral := integral_gaussian_complex hb,
have : measurable_set (Ioi (0:ℝ)) := measurable_set_Ioi,
rw [←integral_add_compl this (integrable_cexp_neg_mul_sq hb), compl_Ioi] at full_integral,
suffices : ∫ x:ℝ in Iic 0, cexp (-b * x^2) = ∫ x:ℝ in Ioi 0, cexp (-b * x^2),
{ rw [this, ←mul_two] at f... | lemma | integral_gaussian_complex_Ioi | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"eq_div_iff",
"integrable_cexp_neg_mul_sq",
"integral_gaussian_complex",
"interval_integral.integral_comp_sub_left",
"measurable_set",
"measurable_set_Ioi",
"neg_sq",
"tendsto_nhds_unique",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_gaussian_Ioi (b : ℝ) : ∫ x in Ioi 0, exp (-b * x^2) = sqrt (π / b) / 2 | begin
rcases le_or_lt b 0 with hb|hb,
{ rw [integral_undef, sqrt_eq_zero_of_nonpos, zero_div],
exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb,
rwa [←integrable_on, integrable_on_Ioi_exp_neg_mul_sq_iff, not_lt] },
rw [←of_real_inj, ←integral_of_real],
convert integral_gaussian_complex_Ioi (by rwa of_r... | lemma | integral_gaussian_Ioi | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"div_nonpos_of_nonneg_of_nonpos",
"div_pos",
"exp",
"integrable_on_Ioi_exp_neg_mul_sq_iff",
"integral_gaussian_complex_Ioi",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.Gamma_one_half_eq : real.Gamma (1 / 2) = sqrt π | begin
rw [Gamma_eq_integral one_half_pos, ←integral_comp_rpow_Ioi_of_pos zero_lt_two],
convert congr_arg (λ x:ℝ, 2 * x) (integral_gaussian_Ioi 1),
{ rw ←integral_mul_left,
refine set_integral_congr measurable_set_Ioi (λ x hx, _),
dsimp only,
have : (x ^ (2:ℝ)) ^ (1 / (2:ℝ) - 1) = x⁻¹,
{ rw ←rpow_m... | lemma | real.Gamma_one_half_eq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"div_one",
"integral_gaussian_Ioi",
"measurable_set_Ioi",
"mul_comm",
"mul_div_cancel",
"one_half_pos",
"real.Gamma",
"ring",
"smul_eq_mul",
"two_ne_zero'",
"zero_lt_two"
] | The special-value formula `Γ(1/2) = √π`, which is equivalent to the Gaussian integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.Gamma_one_half_eq : complex.Gamma (1 / 2) = π ^ (1 / 2 : ℂ) | begin
convert congr_arg coe real.Gamma_one_half_eq,
{ simpa only [one_div, of_real_inv, of_real_bit0] using Gamma_of_real (1 / 2)},
{ rw [sqrt_eq_rpow, of_real_cpow pi_pos.le, of_real_div, of_real_bit0, of_real_one] }
end | lemma | complex.Gamma_one_half_eq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.Gamma",
"one_div",
"real.Gamma_one_half_eq"
] | The special-value formula `Γ(1/2) = √π`, which is equivalent to the Gaussian integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vertical_integral (b : ℂ) (c T : ℝ) : ℂ | ∫ (y : ℝ) in 0..c, I * (cexp (-b * (T + y * I) ^ 2) - cexp (-b * (T - y * I) ^ 2)) | def | gaussian_fourier.vertical_integral | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [] | The integral of the Gaussian function over the vertical edges of a rectangle
with vertices at `(±T, 0)` and `(±T, c)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)) | begin
rw [complex.norm_eq_abs, complex.abs_exp, neg_mul, neg_re, ←re_add_im b],
simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, of_real_re, of_real_im, I_re, I_im],
ring_nf,
end | lemma | gaussian_fourier.norm_cexp_neg_mul_sq_add_mul_I | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.abs_exp",
"complex.norm_eq_abs",
"exp",
"neg_mul"
] | Explicit formula for the norm of the Gaussian function along the vertical
edges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_cexp_neg_mul_sq_add_mul_I' (hb : b.re ≠ 0) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ =
exp (-(b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re))) | begin
have : (b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2) =
b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re),
{ field_simp, ring },
rw [norm_cexp_neg_mul_sq_add_mul_I, this],
end | lemma | gaussian_fourier.norm_cexp_neg_mul_sq_add_mul_I' | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"exp",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vertical_integral_norm_le (hb : 0 < b.re) (c : ℝ) {T : ℝ} (hT : 0 ≤ T) :
‖vertical_integral b c T‖
≤ 2 * |c| * exp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2)) | begin
-- first get uniform bound for integrand
have vert_norm_bound : ∀ {T : ℝ}, 0 ≤ T → ∀ {c y : ℝ}, |y| ≤ |c| →
‖cexp (-b * (T + y * I) ^ 2)‖ ≤ exp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2)),
{ intros T hT c y hy,
rw [norm_cexp_neg_mul_sq_add_mul_I b, exp_le_exp, neg_le_neg_iff],
refine... | lemma | gaussian_fourier.vertical_integral_norm_le | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"abs_mul",
"abs_nonneg",
"abs_of_neg",
"abs_of_nonneg",
"abs_of_nonpos",
"abs_of_pos",
"complex.norm_eq_abs",
"exp",
"interval_integral.norm_integral_le_of_norm_le_const",
"le_abs_self",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"neg_mul",
"nor... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_vertical_integral (hb : 0 < b.re) (c : ℝ) :
tendsto (vertical_integral b c) at_top (𝓝 0) | begin
-- complete proof using squeeze theorem:
rw tendsto_zero_iff_norm_tendsto_zero,
refine tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds _
(eventually_of_forall (λ _, norm_nonneg _))
((eventually_ge_at_top (0:ℝ)).mp (eventually_of_forall
(λ T hT, vertical_integral_norm_le hb c hT))... | lemma | gaussian_fourier.tendsto_vertical_integral | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"ring",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
integrable (λ (x : ℝ), cexp (-b * (x + c * I) ^ 2)) | begin
refine ⟨(complex.continuous_exp.comp (continuous_const.mul ((continuous_of_real.add
continuous_const).pow 2))).ae_strongly_measurable, _⟩,
rw ←has_finite_integral_norm_iff,
simp_rw [norm_cexp_neg_mul_sq_add_mul_I' hb.ne', neg_sub _ (c ^ 2 * _),
sub_eq_add_neg _ (b.re * _), real.exp_add],
suffices ... | lemma | gaussian_fourier.integrable_cexp_neg_mul_sq_add_real_mul_I | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"continuous_const",
"exp",
"integrable_exp_neg_mul_sq",
"real.exp_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
∫ (x : ℝ), cexp (-b * (x + c * I) ^ 2) = (π / b) ^ (1 / 2 : ℂ) | begin
refine tendsto_nhds_unique (interval_integral_tendsto_integral
(integrable_cexp_neg_mul_sq_add_real_mul_I hb c) tendsto_neg_at_top_at_bot tendsto_id) _,
set I₁ := (λ T, ∫ (x : ℝ) in -T..T, cexp (-b * (x + c * I) ^ 2)) with HI₁,
let I₂ := λ (T : ℝ), ∫ (x : ℝ) in -T..T, cexp (-b * x ^ 2),
let I₄ := λ (T... | lemma | gaussian_fourier.integral_cexp_neg_mul_sq_add_real_mul_I | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"algebra.id.smul_eq_mul",
"continuity",
"continuous.interval_integrable",
"differentiable.const_mul",
"differentiable.differentiable_on",
"differentiable_pow",
"integrable_cexp_neg_mul_sq",
"interval_integral.integral_sub",
"mul_one",
"mul_zero",
"tendsto_nhds_unique",
"tsub_zero",
"zero_mul... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.integral_cexp_neg_mul_sq_add_const (hb : 0 < b.re) (c : ℂ) :
∫ (x : ℝ), cexp (-b * (x + c) ^ 2) = (π / b) ^ (1 / 2 : ℂ) | begin
rw ←re_add_im c,
simp_rw [←add_assoc, ←of_real_add],
rw integral_add_right_eq_self (λ(x : ℝ), cexp (-b * (↑x + ↑(c.im) * I) ^ 2)),
{ apply integral_cexp_neg_mul_sq_add_real_mul_I hb },
{ apply_instance },
end | lemma | integral_cexp_neg_mul_sq_add_const | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.fourier_transform_gaussian (hb : 0 < b.re) (t : ℂ) :
∫ (x : ℝ), cexp (I * t * x) * cexp (-b * x ^ 2) = cexp (-t^2 / (4 * b)) * (π / b) ^ (1 / 2 : ℂ) | begin
have : b ≠ 0,
{ contrapose! hb, rw [hb, zero_re] },
simp_rw [←complex.exp_add],
have : ∀ (x : ℂ), I * t * x + (-b * x ^ 2) = -t ^ 2 / (4 * b) + -b * (x + (-I * t / 2 / b)) ^ 2,
{ intro x,
ring_nf SOP,
rw I_sq,
field_simp, ring },
simp_rw [this, complex.exp_add, integral_mul_left, integral_... | lemma | fourier_transform_gaussian | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.exp_add",
"integral_cexp_neg_mul_sq_add_const",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.fourier_transform_gaussian_pi (hb : 0 < b.re) :
𝓕 (λ x : ℝ, cexp (-π * b * x ^ 2)) = λ t : ℝ, 1 / b ^ (1 / 2 : ℂ) * cexp (-π / b * t ^ 2) | begin
ext1 t,
simp_rw [fourier_integral_eq_integral_exp_smul, smul_eq_mul],
have h1 : 0 < re (π * b) := by { rw of_real_mul_re, exact mul_pos pi_pos hb },
have h2 : b ≠ 0 := by { contrapose! hb, rw [hb, zero_re], },
convert _root_.fourier_transform_gaussian h1 (-2 * π * t) using 1,
{ congr' 1 with x:1,
... | lemma | fourier_transform_gaussian_pi | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"div_self",
"mul_comm",
"not_and_distrib",
"one_div",
"ring",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact {a : ℝ} (ha : 0 < a) (s : ℝ) :
tendsto (λ x : ℝ, |x| ^ s * rexp (-a * x ^ 2)) (cocompact ℝ) (𝓝 0) | begin
conv in (rexp _) { rw ←sq_abs },
rw [cocompact_eq, ←comap_abs_at_top,
@tendsto_comap'_iff _ _ _ (λ y, y ^ s * rexp (-a * y ^ 2)) _ _ _
(mem_at_top_sets.mpr ⟨0, λ b hb, ⟨b, abs_of_nonneg hb⟩⟩)],
exact (rpow_mul_exp_neg_mul_sq_is_o_exp_neg ha s).tendsto_zero_of_tendsto
(tendsto_exp_at_bot.comp $ t... | lemma | tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"abs_of_nonneg",
"one_half_pos",
"rpow_mul_exp_neg_mul_sq_is_o_exp_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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