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
value |
|---|---|---|---|---|---|---|---|---|---|---|
circle_integral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable
{c : ℂ} {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : set ℂ} (hs : s.countable)
(hc : continuous_on f (closed_ball c R \ ball c r))
(hd : ∀ z ∈ ball c R \ closed_ball c r \ s, differentiable_at ℂ f z) :
∮ z in C(c, R), (z ... | begin
/- We apply the previous lemma to `λ z, f (c + exp z)` on the rectangle
`[log r, log R] × [0, 2 * π]`. -/
set A := closed_ball c R \ ball c r,
obtain ⟨a, rfl⟩ : ∃ a, real.exp a = r, from ⟨real.log r, real.exp_log h0⟩,
obtain ⟨b, rfl⟩ : ∃ b, real.exp b = R, from ⟨real.log R, real.exp_log (h0.trans_le hle... | lemma | complex.circle_integral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"circle_integral",
"circle_map",
"circle_map_ne_center",
"circle_map_sub_center",
"continuous_on",
"countable",
"deriv_circle_map",
"differentiable",
"differentiable_at",
"div_eq_mul_inv",
"exp",
"exp_add",
"mul_div_cancel_left",
"real.exp",
"real.exp_le_exp",
"real.exp_log",
"real.e... | If `f : ℂ → E` is continuous the closed annulus `r ≤ ‖z - c‖ ≤ R`, `0 < r ≤ R`, and is complex
differentiable at all but countably many points of its interior, then the integrals of
`f z / (z - c)` (formally, `(z - c)⁻¹ • f z`) over the circles `‖z - c‖ = r` and `‖z - c‖ = R` are
equal to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_eq_of_differentiable_on_annulus_off_countable
{c : ℂ} {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : set ℂ} (hs : s.countable)
(hc : continuous_on f (closed_ball c R \ ball c r))
(hd : ∀ z ∈ ball c R \ closed_ball c r \ s, differentiable_at ℂ f z) :
∮ z in C(c, R), f z = ∮ z in C(c, r), f... | calc ∮ z in C(c, R), f z = ∮ z in C(c, R), (z - c)⁻¹ • (z - c) • f z :
(circle_integral.integral_sub_inv_smul_sub_smul _ _ _ _).symm
... = ∮ z in C(c, r), (z - c)⁻¹ • (z - c) • f z :
circle_integral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable h0 hle hs
((continuous_on_id.sub continuous_on_... | lemma | complex.circle_integral_eq_of_differentiable_on_annulus_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"circle_integral.integral_sub_inv_smul_sub_smul",
"continuous_on",
"continuous_on_const",
"differentiable_at"
] | **Cauchy-Goursat theorem** for an annulus. If `f : ℂ → E` is continuous on the closed annulus
`r ≤ ‖z - c‖ ≤ R`, `0 < r ≤ R`, and is complex differentiable at all but countably many points of
its interior, then the integrals of `f` over the circles `‖z - c‖ = r` and `‖z - c‖ = R` are equal
to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto
{c : ℂ} {R : ℝ} (h0 : 0 < R) {f : ℂ → E} {y : E} {s : set ℂ} (hs : s.countable)
(hc : continuous_on f (closed_ball c R \ {c}))
(hd : ∀ z ∈ ball c R \ {c} \ s, differentiable_at ℂ f z) (hy : tendsto f (𝓝[{c}ᶜ] c) (𝓝 y)) :
∮ z in ... | begin
rw [← sub_eq_zero, ← norm_le_zero_iff],
refine le_of_forall_le_of_dense (λ ε ε0, _),
obtain ⟨δ, δ0, hδ⟩ :
∃ δ > (0 : ℝ), ∀ z ∈ closed_ball c δ \ {c}, dist (f z) y < ε / (2 * π),
from ((nhds_within_has_basis nhds_basis_closed_ball _).tendsto_iff nhds_basis_ball).1 hy _
(div_pos ε0 real.two_pi_p... | lemma | complex.circle_integral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"circle_integrable",
"circle_integral.integral_sub",
"circle_integral.norm_integral_le_of_norm_le_const",
"continuous_on",
"continuous_on_const",
"differentiable_at",
"dist_self",
"div_pos",
"le_of_forall_le_of_dense",
"mul_le_mul_of_nonneg_left",
"ne_of_mem_of_not_mem",
"nhds_within_has_basis... | **Cauchy integral formula** for the value at the center of a disc. If `f` is continuous on a
punctured closed disc of radius `R`, is differentiable at all but countably many points of the
interior of this disc, and has a limit `y` at the center of the disc, then the integral
$\oint_{‖z-c‖=R} \frac{f(z)}{z-c}\,dz$ is eq... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_sub_center_inv_smul_of_differentiable_on_off_countable {R : ℝ} (h0 : 0 < R)
{f : ℂ → E} {c : ℂ} {s : set ℂ} (hs : s.countable)
(hc : continuous_on f (closed_ball c R)) (hd : ∀ z ∈ ball c R \ s, differentiable_at ℂ f z) :
∮ z in C(c, R), (z - c)⁻¹ • f z = (2 * π * I : ℂ) • f c | circle_integral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto h0 hs
(hc.mono $ diff_subset _ _) (λ z hz, hd z ⟨hz.1.1, hz.2⟩)
(hc.continuous_at $ closed_ball_mem_nhds _ h0).continuous_within_at | lemma | complex.circle_integral_sub_center_inv_smul_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_on",
"continuous_within_at",
"differentiable_at"
] | **Cauchy integral formula** for the value at the center of a disc. If `f : ℂ → E` is continuous
on a closed disc of radius `R` and is complex differentiable at all but countably many points of its
interior, then the integral $\oint_{|z-c|=R} \frac{f(z)}{z-c}\,dz$ is equal to $2πiy`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_eq_zero_of_differentiable_on_off_countable {R : ℝ} (h0 : 0 ≤ R) {f : ℂ → E}
{c : ℂ} {s : set ℂ} (hs : s.countable) (hc : continuous_on f (closed_ball c R))
(hd : ∀ z ∈ ball c R \ s, differentiable_at ℂ f z) :
∮ z in C(c, R), f z = 0 | begin
rcases h0.eq_or_lt with rfl|h0, { apply circle_integral.integral_radius_zero },
calc ∮ z in C(c, R), f z = ∮ z in C(c, R), (z - c)⁻¹ • (z - c) • f z :
(circle_integral.integral_sub_inv_smul_sub_smul _ _ _ _).symm
... = (2 * ↑π * I : ℂ) • (c - c) • f c :
circle_integral_sub_center_inv_smul_of_differe... | lemma | complex.circle_integral_eq_zero_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"circle_integral.integral_radius_zero",
"circle_integral.integral_sub_inv_smul_sub_smul",
"continuous_on",
"continuous_on_const",
"differentiable_at",
"smul_zero",
"zero_smul"
] | **Cauchy-Goursat theorem** for a disk: if `f : ℂ → E` is continuous on a closed disk
`{z | ‖z - c‖ ≤ R}` and is complex differentiable at all but countably many points of its interior,
then the integral $\oint_{|z-c|=R}f(z)\,dz$ equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_sub_inv_smul_of_differentiable_on_off_countable_aux {R : ℝ} {c w : ℂ}
{f : ℂ → E} {s : set ℂ} (hs : s.countable) (hw : w ∈ ball c R \ s)
(hc : continuous_on f (closed_ball c R)) (hd : ∀ x ∈ ball c R \ s, differentiable_at ℂ f x) :
∮ z in C(c, R), (z - w)⁻¹ • f z = (2 * π * I : ℂ) • f w | begin
have hR : 0 < R := dist_nonneg.trans_lt hw.1,
set F : ℂ → E := dslope f w,
have hws : (insert w s).countable := hs.insert w,
have hnhds : closed_ball c R ∈ 𝓝 w, from closed_ball_mem_nhds_of_mem hw.1,
have hcF : continuous_on F (closed_ball c R),
from (continuous_on_dslope $ closed_ball_mem_nhds_of_... | lemma | complex.circle_integral_sub_inv_smul_of_differentiable_on_off_countable_aux | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"circle_integrable",
"circle_integral.integral_congr",
"circle_integral.integral_smul_const",
"circle_integral.integral_sub",
"circle_integral.integral_sub_inv_of_mem_ball",
"continuous_on",
"continuous_on_const",
"continuous_on_dslope",
"countable",
"differentiable_at",
"differentiable_at_dslop... | An auxiliary lemma for
`complex.circle_integral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_pi_I_inv_smul_circle_integral_sub_inv_smul_of_differentiable_on_off_countable
{R : ℝ} {c w : ℂ} {f : ℂ → E} {s : set ℂ} (hs : s.countable) (hw : w ∈ ball c R)
(hc : continuous_on f (closed_ball c R)) (hd : ∀ x ∈ ball c R \ s, differentiable_at ℂ f x) :
(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z = f w | begin
have hR : 0 < R := dist_nonneg.trans_lt hw,
suffices : w ∈ closure (ball c R \ s),
{ lift R to ℝ≥0 using hR.le,
have A : continuous_at (λ w, (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w,
{ have := has_fpower_series_on_cauchy_integral
((hc.mono sphere_subset_closed_ball).circle_integr... | lemma | complex.two_pi_I_inv_smul_circle_integral_sub_inv_smul_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"cardinal.aleph_0_lt_continuum",
"cardinal.le_aleph_0_iff_set_countable",
"cardinal.mk_Ioo_real",
"circle_integrable",
"closure",
"continuous_at",
"continuous_on",
"countable",
"differentiable_at",
"has_fpower_series_on_cauchy_integral",
"inv_smul_smul₀",
"lift",
"metric.emetric_ball_nnreal"... | **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_sub_inv_smul_of_differentiable_on_off_countable
{R : ℝ} {c w : ℂ} {f : ℂ → E} {s : set ℂ} (hs : s.countable) (hw : w ∈ ball c R)
(hc : continuous_on f (closed_ball c R)) (hd : ∀ x ∈ ball c R \ s, differentiable_at ℂ f x) :
∮ z in C(c, R), (z - w)⁻¹ • f z = (2 * π * I : ℂ) • f w | by { rw [← two_pi_I_inv_smul_circle_integral_sub_inv_smul_of_differentiable_on_off_countable
hs hw hc hd, smul_inv_smul₀], simp [real.pi_ne_zero, I_ne_zero] } | lemma | complex.circle_integral_sub_inv_smul_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_on",
"differentiable_at",
"real.pi_ne_zero",
"smul_inv_smul₀"
] | **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.diff_cont_on_cl.circle_integral_sub_inv_smul {R : ℝ} {c w : ℂ} {f : ℂ → E}
(h : diff_cont_on_cl ℂ f (ball c R)) (hw : w ∈ ball c R) :
∮ z in C(c, R), (z - w)⁻¹ • f z = (2 * π * I : ℂ) • f w | circle_integral_sub_inv_smul_of_differentiable_on_off_countable countable_empty hw
h.continuous_on_ball $ λ x hx, h.differentiable_at is_open_ball hx.1 | lemma | diff_cont_on_cl.circle_integral_sub_inv_smul | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"diff_cont_on_cl"
] | **Cauchy integral formula**: if `f : ℂ → E` is complex differentiable on an open disc and is
continuous on its closure, then for any `w` in this open ball we have
$\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.diff_cont_on_cl.two_pi_I_inv_smul_circle_integral_sub_inv_smul {R : ℝ} {c w : ℂ}
{f : ℂ → E} (hf : diff_cont_on_cl ℂ f (ball c R)) (hw : w ∈ ball c R) :
(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z = f w | begin
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (nonempty_of_mem hw).ne_empty),
refine two_pi_I_inv_smul_circle_integral_sub_inv_smul_of_differentiable_on_off_countable
countable_empty hw _ _,
{ simpa only [closure_ball c hR.ne.symm] using hf.continuous_on },
{ simpa only [diff_empty] using λ z hz,... | lemma | diff_cont_on_cl.two_pi_I_inv_smul_circle_integral_sub_inv_smul | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"closure_ball",
"diff_cont_on_cl"
] | **Cauchy integral formula**: if `f : ℂ → E` is complex differentiable on an open disc and is
continuous on its closure, then for any `w` in this open ball we have
$\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.differentiable_on.circle_integral_sub_inv_smul {R : ℝ} {c w : ℂ} {f : ℂ → E}
(hd : differentiable_on ℂ f (closed_ball c R)) (hw : w ∈ ball c R) :
∮ z in C(c, R), (z - w)⁻¹ • f z = (2 * π * I : ℂ) • f w | (hd.mono closure_ball_subset_closed_ball).diff_cont_on_cl.circle_integral_sub_inv_smul hw | lemma | differentiable_on.circle_integral_sub_inv_smul | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"diff_cont_on_cl.circle_integral_sub_inv_smul",
"differentiable_on"
] | **Cauchy integral formula**: if `f : ℂ → E` is complex differentiable on a closed disc of radius
`R`, then for any `w` in its interior we have $\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle_integral_div_sub_of_differentiable_on_off_countable {R : ℝ} {c w : ℂ} {s : set ℂ}
(hs : s.countable) (hw : w ∈ ball c R) {f : ℂ → ℂ} (hc : continuous_on f (closed_ball c R))
(hd : ∀ z ∈ ball c R \ s, differentiable_at ℂ f z) :
∮ z in C(c, R), f z / (z - w) = 2 * π * I * f w | by simpa only [smul_eq_mul, div_eq_inv_mul]
using circle_integral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd | lemma | complex.circle_integral_div_sub_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_on",
"differentiable_at",
"div_eq_inv_mul",
"smul_eq_mul"
] | **Cauchy integral formula**: if `f : ℂ → ℂ` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\oint_{|z-c|=R}\frac{f(z)}{z-w}dz=2\pi i\,f(w)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball_of_differentiable_off_countable {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
{s : set ℂ} (hs : s.countable) (hc : continuous_on f (closed_ball c R))
(hd : ∀ z ∈ ball c R \ s, differentiable_at ℂ f z) (hR : 0 < R) :
has_fpower_series_on_ball f (cauchy_power_series f c R) c R | { r_le := le_radius_cauchy_power_series _ _ _,
r_pos := ennreal.coe_pos.2 hR,
has_sum := λ w hw,
begin
have hw' : c + w ∈ ball c R,
by simpa only [add_mem_ball_iff_norm, ← coe_nnnorm, mem_emetric_ball_zero_iff,
nnreal.coe_lt_coe, ennreal.coe_lt_coe] using hw,
rw ← two_pi_I_inv_smul... | lemma | complex.has_fpower_series_on_ball_of_differentiable_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"cauchy_power_series",
"circle_integrable",
"continuous_on",
"differentiable_at",
"ennreal.coe_lt_coe",
"has_fpower_series_on_ball",
"has_fpower_series_on_cauchy_integral",
"has_sum",
"le_radius_cauchy_power_series",
"nnreal.coe_lt_coe"
] | If `f : ℂ → E` is continuous on a closed ball of positive radius and is differentiable at all
but countably many points of the corresponding open ball, then it is analytic on the open ball with
coefficients of the power series given by Cauchy integral formulas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.diff_cont_on_cl.has_fpower_series_on_ball {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
(hf : diff_cont_on_cl ℂ f (ball c R)) (hR : 0 < R) :
has_fpower_series_on_ball f (cauchy_power_series f c R) c R | has_fpower_series_on_ball_of_differentiable_off_countable countable_empty hf.continuous_on_ball
(λ z hz, hf.differentiable_at is_open_ball hz.1) hR | lemma | diff_cont_on_cl.has_fpower_series_on_ball | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"cauchy_power_series",
"diff_cont_on_cl",
"has_fpower_series_on_ball"
] | If `f : ℂ → E` is complex differentiable on an open disc of positive radius and is continuous
on its closure, then it is analytic on the open disc with coefficients of the power series given by
Cauchy integral formulas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.differentiable_on.has_fpower_series_on_ball {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
(hd : differentiable_on ℂ f (closed_ball c R)) (hR : 0 < R) :
has_fpower_series_on_ball f (cauchy_power_series f c R) c R | (hd.mono closure_ball_subset_closed_ball).diff_cont_on_cl.has_fpower_series_on_ball hR | lemma | differentiable_on.has_fpower_series_on_ball | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"cauchy_power_series",
"diff_cont_on_cl.has_fpower_series_on_ball",
"differentiable_on",
"has_fpower_series_on_ball"
] | If `f : ℂ → E` is complex differentiable on a closed disc of positive radius, then it is
analytic on the corresponding open disc, and the coefficients of the power series are given by
Cauchy integral formulas. See also
`complex.has_fpower_series_on_ball_of_differentiable_off_countable` for a version of this lemma with
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.differentiable_on.analytic_at {s : set ℂ} {f : ℂ → E} {z : ℂ}
(hd : differentiable_on ℂ f s) (hz : s ∈ 𝓝 z) : analytic_at ℂ f z | begin
rcases nhds_basis_closed_ball.mem_iff.1 hz with ⟨R, hR0, hRs⟩,
lift R to ℝ≥0 using hR0.le,
exact ((hd.mono hRs).has_fpower_series_on_ball hR0).analytic_at
end | lemma | differentiable_on.analytic_at | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"analytic_at",
"differentiable_on",
"has_fpower_series_on_ball",
"lift"
] | If `f : ℂ → E` is complex differentiable on some set `s`, then it is analytic at any point `z`
such that `s ∈ 𝓝 z` (equivalently, `z ∈ interior s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.differentiable_on.analytic_on {s : set ℂ} {f : ℂ → E} (hd : differentiable_on ℂ f s)
(hs : is_open s) : analytic_on ℂ f s | λ z hz, hd.analytic_at (hs.mem_nhds hz) | lemma | differentiable_on.analytic_on | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"analytic_on",
"differentiable_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.differentiable.analytic_at {f : ℂ → E} (hf : differentiable ℂ f) (z : ℂ) :
analytic_at ℂ f z | hf.differentiable_on.analytic_at univ_mem | lemma | differentiable.analytic_at | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"analytic_at",
"differentiable"
] | A complex differentiable function `f : ℂ → E` is analytic at every point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.differentiable.has_fpower_series_on_ball {f : ℂ → E}
(h : differentiable ℂ f) (z : ℂ) {R : ℝ≥0} (hR : 0 < R) :
has_fpower_series_on_ball f (cauchy_power_series f z R) z ∞ | (h.differentiable_on.has_fpower_series_on_ball hR).r_eq_top_of_exists $ λ r hr,
⟨_, h.differentiable_on.has_fpower_series_on_ball hr⟩ | lemma | differentiable.has_fpower_series_on_ball | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"cauchy_power_series",
"differentiable",
"has_fpower_series_on_ball"
] | When `f : ℂ → E` is differentiable, the `cauchy_power_series f z R` represents `f` as a power
series centered at `z` in the entirety of `ℂ`, regardless of `R : ℝ≥0`, with `0 < R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle : submonoid ℂ | submonoid.unit_sphere ℂ | def | circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"submonoid",
"submonoid.unit_sphere"
] | The unit circle in `ℂ`, here given the structure of a submonoid of `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_circle_iff_abs {z : ℂ} : z ∈ circle ↔ abs z = 1 | mem_sphere_zero_iff_norm | lemma | mem_circle_iff_abs | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circle_def : ↑circle = {z : ℂ | abs z = 1} | set.ext $ λ z, mem_circle_iff_abs | lemma | circle_def | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"mem_circle_iff_abs",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_coe_circle (z : circle) : abs z = 1 | mem_circle_iff_abs.mp z.2 | lemma | abs_coe_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_circle_iff_norm_sq {z : ℂ} : z ∈ circle ↔ norm_sq z = 1 | by simp [complex.abs] | lemma | mem_circle_iff_norm_sq | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle",
"complex.abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_eq_of_mem_circle (z : circle) : norm_sq z = 1 | by simp [norm_sq_eq_abs] | lemma | norm_sq_eq_of_mem_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mem_circle (z : circle) : (z:ℂ) ≠ 0 | ne_zero_of_mem_unit_sphere z | lemma | ne_zero_of_mem_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_circle (z : circle) : ↑(z⁻¹) = (z : ℂ)⁻¹ | rfl | lemma | coe_inv_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_circle_eq_conj (z : circle) : ↑(z⁻¹) = conj (z : ℂ) | by rw [coe_inv_circle, inv_def, norm_sq_eq_of_mem_circle, inv_one, of_real_one, mul_one] | lemma | coe_inv_circle_eq_conj | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle",
"coe_inv_circle",
"inv_one",
"mul_one",
"norm_sq_eq_of_mem_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div_circle (z w : circle) : ↑(z / w) = (z:ℂ) / w | circle.subtype.map_div z w | lemma | coe_div_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circle.to_units : circle →* units ℂ | unit_sphere_to_units ℂ | def | circle.to_units | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle",
"unit_sphere_to_units",
"units"
] | The elements of the circle embed into the units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
circle.to_units_apply (z : circle) :
circle.to_units z = units.mk0 z (ne_zero_of_mem_circle z) | rfl | lemma | circle.to_units_apply | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle",
"circle.to_units",
"ne_zero_of_mem_circle",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circle.of_conj_div_self (z : ℂ) (hz : z ≠ 0) : circle | ⟨conj z / z, mem_circle_iff_abs.2 $ by rw [map_div₀, abs_conj, div_self (complex.abs.ne_zero hz)]⟩ | def | circle.of_conj_div_self | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle",
"div_self",
"map_div₀"
] | If `z` is a nonzero complex number, then `conj z / z` belongs to the unit circle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_map_circle : C(ℝ, circle) | { to_fun := λ t, ⟨exp (t * I), by simp [exp_mul_I, abs_cos_add_sin_mul_I]⟩ } | def | exp_map_circle | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"circle"
] | The map `λ t, exp (t * I)` from `ℝ` to the unit circle in `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_map_circle_apply (t : ℝ) : ↑(exp_map_circle t) = complex.exp (t * complex.I) | rfl | lemma | exp_map_circle_apply | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"complex.I",
"complex.exp",
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_zero : exp_map_circle 0 = 1 | subtype.ext $ by rw [exp_map_circle_apply, of_real_zero, zero_mul, exp_zero, submonoid.coe_one] | lemma | exp_map_circle_zero | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"exp_map_circle",
"exp_map_circle_apply",
"exp_zero",
"submonoid.coe_one",
"subtype.ext",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_add (x y : ℝ) :
exp_map_circle (x + y) = exp_map_circle x * exp_map_circle y | subtype.ext $ by simp only [exp_map_circle_apply, submonoid.coe_mul, of_real_add, add_mul,
complex.exp_add] | lemma | exp_map_circle_add | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"complex.exp_add",
"exp_map_circle",
"exp_map_circle_apply",
"submonoid.coe_mul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_hom : ℝ →+ (additive circle) | { to_fun := additive.of_mul ∘ exp_map_circle,
map_zero' := exp_map_circle_zero,
map_add' := exp_map_circle_add } | def | exp_map_circle_hom | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"additive",
"additive.of_mul",
"circle",
"exp_map_circle",
"exp_map_circle_add",
"exp_map_circle_zero"
] | The map `λ t, exp (t * I)` from `ℝ` to the unit circle in `ℂ`, considered as a homomorphism of
groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_map_circle_sub (x y : ℝ) :
exp_map_circle (x - y) = exp_map_circle x / exp_map_circle y | exp_map_circle_hom.map_sub x y | lemma | exp_map_circle_sub | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_neg (x : ℝ) : exp_map_circle (-x) = (exp_map_circle x)⁻¹ | exp_map_circle_hom.map_neg x | lemma | exp_map_circle_neg | analysis.complex | src/analysis/complex/circle.lean | [
"analysis.special_functions.exp",
"topology.continuous_function.basic",
"analysis.normed.field.unit_ball"
] | [
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_conj : is_conformal_map (conj_lie : ℂ →L[ℝ] ℂ) | conj_lie.to_linear_isometry.is_conformal_map | lemma | is_conformal_map_conj | analysis.complex | src/analysis/complex/conformal.lean | [
"analysis.complex.isometry",
"analysis.normed_space.conformal_linear_map",
"analysis.normed_space.finite_dimension"
] | [
"is_conformal_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_complex_linear {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
is_conformal_map (map.restrict_scalars ℝ) | begin
have minor₁ : ‖map 1‖ ≠ 0,
{ simpa only [ext_ring_iff, ne.def, norm_eq_zero] using nonzero},
refine ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • map, _⟩, _⟩,
{ intros x,
simp only [linear_map.smul_apply],
have : x = x • 1 := by rw [smul_eq_mul, mul_one],
nth_rewrite 0 [this],
rw [_root_.coe_coe map, lin... | lemma | is_conformal_map_complex_linear | analysis.complex | src/analysis/complex/conformal.lean | [
"analysis.complex.isometry",
"analysis.normed_space.conformal_linear_map",
"analysis.normed_space.finite_dimension"
] | [
"continuous_linear_map.coe_coe",
"is_conformal_map",
"linear_isometry.coe_mk",
"linear_isometry.coe_to_continuous_linear_map",
"linear_map.coe_coe_is_scalar_tower",
"linear_map.smul_apply",
"mul_one",
"norm_eq_zero",
"norm_inv",
"norm_norm",
"norm_smul",
"one_mul",
"pi.smul_apply",
"smul_e... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_complex_linear_conj
{map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
is_conformal_map ((map.restrict_scalars ℝ).comp (conj_cle : ℂ →L[ℝ] ℂ)) | (is_conformal_map_complex_linear nonzero).comp is_conformal_map_conj | lemma | is_conformal_map_complex_linear_conj | analysis.complex | src/analysis/complex/conformal.lean | [
"analysis.complex.isometry",
"analysis.normed_space.conformal_linear_map",
"analysis.normed_space.finite_dimension"
] | [
"is_conformal_map",
"is_conformal_map_complex_linear",
"is_conformal_map_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map.is_complex_or_conj_linear (h : is_conformal_map g) :
(∃ (map : ℂ →L[ℂ] ℂ), map.restrict_scalars ℝ = g) ∨
(∃ (map : ℂ →L[ℂ] ℂ), map.restrict_scalars ℝ = g ∘L ↑conj_cle) | begin
rcases h with ⟨c, hc, li, rfl⟩,
obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.to_linear_isometry = li,
from ⟨li.to_linear_isometry_equiv rfl, by { ext1, refl }⟩,
rcases linear_isometry_complex li with ⟨a, rfl|rfl⟩,
-- let rot := c • (a : ℂ) • continuous_linear_map.id ℂ ℂ,
{ refine or.inl ⟨c • (a : ℂ) •... | lemma | is_conformal_map.is_complex_or_conj_linear | analysis.complex | src/analysis/complex/conformal.lean | [
"analysis.complex.isometry",
"analysis.normed_space.conformal_linear_map",
"analysis.normed_space.finite_dimension"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_map.id",
"is_conformal_map",
"linear_isometry.coe_to_continuous_linear_map",
"linear_isometry_complex",
"linear_isometry_equiv.coe_to_linear_isometry",
"linear_isometry_equiv.trans_apply",
"rotation_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_iff_is_complex_or_conj_linear:
is_conformal_map g ↔
((∃ (map : ℂ →L[ℂ] ℂ), map.restrict_scalars ℝ = g) ∨
(∃ (map : ℂ →L[ℂ] ℂ), map.restrict_scalars ℝ = g ∘L ↑conj_cle)) ∧ g ≠ 0 | begin
split,
{ exact λ h, ⟨h.is_complex_or_conj_linear, h.ne_zero⟩, },
{ rintros ⟨⟨map, rfl⟩ | ⟨map, hmap⟩, h₂⟩,
{ refine is_conformal_map_complex_linear _,
contrapose! h₂ with w,
simp only [w, restrict_scalars_zero]},
{ have minor₁ : g = (map.restrict_scalars ℝ) ∘L ↑conj_cle,
{ ext1,
... | lemma | is_conformal_map_iff_is_complex_or_conj_linear | analysis.complex | src/analysis/complex/conformal.lean | [
"analysis.complex.isometry",
"analysis.normed_space.conformal_linear_map",
"analysis.normed_space.finite_dimension"
] | [
"continuous_linear_equiv.coe_coe",
"is_conformal_map",
"is_conformal_map_complex_linear",
"is_conformal_map_complex_linear_conj",
"star_ring_end_self_apply"
] | A real continuous linear map on the complex plane is conformal if and only if the map or its
conjugate is complex linear, and the map is nonvanishing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rotation : circle →* (ℂ ≃ₗᵢ[ℝ] ℂ) | { to_fun := λ a,
{ norm_map' := λ x, show |a * x| = |x|, by rw [map_mul, abs_coe_circle, one_mul],
..distrib_mul_action.to_linear_equiv ℝ ℂ a },
map_one' := linear_isometry_equiv.ext $ one_smul _,
map_mul' := λ _ _, linear_isometry_equiv.ext $ mul_smul _ _ } | def | rotation | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"abs_coe_circle",
"circle",
"distrib_mul_action.to_linear_equiv",
"linear_isometry_equiv.ext",
"map_mul",
"one_mul",
"one_smul"
] | An element of the unit circle defines a `linear_isometry_equiv` from `ℂ` to itself, by
rotation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rotation_apply (a : circle) (z : ℂ) : rotation a z = a * z | rfl | lemma | rotation_apply | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"rotation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rotation_symm (a : circle) : (rotation a).symm = rotation a⁻¹ | linear_isometry_equiv.ext $ λ x, rfl | lemma | rotation_symm | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"linear_isometry_equiv.ext",
"rotation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rotation_trans (a b : circle) :
(rotation a).trans (rotation b) = rotation (b * a) | by { ext1, simp } | lemma | rotation_trans | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"rotation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rotation_ne_conj_lie (a : circle) : rotation a ≠ conj_lie | begin
intro h,
have h1 : rotation a 1 = conj 1 := linear_isometry_equiv.congr_fun h 1,
have hI : rotation a I = conj I := linear_isometry_equiv.congr_fun h I,
rw [rotation_apply, ring_hom.map_one, mul_one] at h1,
rw [rotation_apply, conj_I, ← neg_one_mul, mul_left_inj' I_ne_zero, h1, eq_neg_self_iff] at hI,
... | lemma | rotation_ne_conj_lie | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"eq_neg_self_iff",
"linear_isometry_equiv.congr_fun",
"mul_left_inj'",
"mul_one",
"neg_one_mul",
"one_ne_zero",
"ring_hom.map_one",
"rotation",
"rotation_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rotation_of (e : ℂ ≃ₗᵢ[ℝ] ℂ) : circle | ⟨(e 1) / complex.abs (e 1), by simp⟩ | def | rotation_of | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"complex.abs"
] | Takes an element of `ℂ ≃ₗᵢ[ℝ] ℂ` and checks if it is a rotation, returns an element of the
unit circle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rotation_of_rotation (a : circle) : rotation_of (rotation a) = a | subtype.ext $ by simp | lemma | rotation_of_rotation | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"rotation",
"rotation_of",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rotation_injective : function.injective rotation | function.left_inverse.injective rotation_of_rotation | lemma | rotation_injective | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"rotation",
"rotation_of_rotation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.re_apply_eq_re_of_add_conj_eq (f : ℂ →ₗᵢ[ℝ] ℂ)
(h₃ : ∀ z, z + conj z = f z + conj (f z)) (z : ℂ) : (f z).re = z.re | by simpa [ext_iff, add_re, add_im, conj_re, conj_im, ←two_mul,
(show (2 : ℝ) ≠ 0, by simp [two_ne_zero])] using (h₃ z).symm | lemma | linear_isometry.re_apply_eq_re_of_add_conj_eq | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.im_apply_eq_im_or_neg_of_re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ}
(h₂ : ∀ z, (f z).re = z.re) (z : ℂ) :
(f z).im = z.im ∨ (f z).im = -z.im | begin
have h₁ := f.norm_map z,
simp only [complex.abs_def, norm_eq_abs] at h₁,
rwa [real.sqrt_inj (norm_sq_nonneg _) (norm_sq_nonneg _), norm_sq_apply (f z), norm_sq_apply z,
h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁,
end | lemma | linear_isometry.im_apply_eq_im_or_neg_of_re_apply_eq_re | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"complex.abs_def",
"mul_self_eq_mul_self_iff",
"real.sqrt_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
z + conj z = f z + conj (f z) | begin
have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h],
apply_fun λ x, x ^ 2 at this,
simp only [norm_eq_abs, ←norm_sq_eq_abs] at this,
rw [←of_real_inj, ←mul_conj, ←mul_conj] at this,
rw [ring_hom.map_sub, ring_hom.map_sub] at this,
simp only [sub_mul, mul_sub, one_mul, mul_one] at ... | lemma | linear_isometry.im_apply_eq_im | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"linear_isometry.norm_map",
"mul_one",
"one_mul",
"ring_hom.map_one",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re | begin
apply linear_isometry.re_apply_eq_re_of_add_conj_eq,
intro z,
apply linear_isometry.im_apply_eq_im h,
end | lemma | linear_isometry.re_apply_eq_re | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"linear_isometry.im_apply_eq_im",
"linear_isometry.re_apply_eq_re_of_add_conj_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
f = linear_isometry_equiv.refl ℝ ℂ ∨ f = conj_lie | begin
have h0 : f I = I ∨ f I = -I,
{ have : |f I| = 1 := by simpa using f.norm_map complex.I,
simp only [ext_iff, ←and_or_distrib_left, neg_re, I_re, neg_im, neg_zero],
split,
{ rw ←I_re,
exact @linear_isometry.re_apply_eq_re f.to_linear_isometry h I, },
{ apply @linear_isometry.im_apply_eq_i... | lemma | linear_isometry_complex_aux | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"complex.I",
"linear_isometry.im_apply_eq_im_or_neg_of_re_apply_eq_re",
"linear_isometry.re_apply_eq_re",
"linear_isometry_equiv.refl",
"linear_isometry_equiv.to_linear_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
∃ a : circle, f = rotation a ∨ f = conj_lie.trans (rotation a) | begin
let a : circle := ⟨f 1, by simpa using f.norm_map 1⟩,
use a,
have : (f.trans (rotation a).symm) 1 = 1,
{ simpa using rotation_apply a⁻¹ (f 1) },
refine (linear_isometry_complex_aux this).imp (λ h₁, _) (λ h₂, _),
{ simpa using eq_mul_of_inv_mul_eq h₁ },
{ exact eq_mul_of_inv_mul_eq h₂ }
end | lemma | linear_isometry_complex | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"eq_mul_of_inv_mul_eq",
"linear_isometry_complex_aux",
"rotation",
"rotation_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_matrix_rotation (a : circle) :
linear_map.to_matrix basis_one_I basis_one_I (rotation a).to_linear_equiv =
matrix.plane_conformal_matrix (re a) (im a) (by simp [pow_two, ←norm_sq_apply]) | begin
ext i j,
simp [linear_map.to_matrix_apply],
fin_cases i; fin_cases j; simp
end | lemma | to_matrix_rotation | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"linear_map.to_matrix",
"linear_map.to_matrix_apply",
"matrix.plane_conformal_matrix",
"pow_two",
"rotation"
] | The matrix representation of `rotation a` is equal to the conformal matrix
`!![re a, -im a; im a, re a]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_rotation (a : circle) : ((rotation a).to_linear_equiv : ℂ →ₗ[ℝ] ℂ).det = 1 | begin
rw [←linear_map.det_to_matrix basis_one_I, to_matrix_rotation, matrix.det_fin_two],
simp [←norm_sq_apply]
end | lemma | det_rotation | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"matrix.det_fin_two",
"rotation",
"to_matrix_rotation"
] | The determinant of `rotation` (as a linear map) is equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_det_rotation (a : circle) : (rotation a).to_linear_equiv.det = 1 | by rw [←units.eq_iff, linear_equiv.coe_det, det_rotation, units.coe_one] | lemma | linear_equiv_det_rotation | analysis.complex | src/analysis/complex/isometry.lean | [
"analysis.complex.circle",
"linear_algebra.determinant",
"linear_algebra.matrix.general_linear_group"
] | [
"circle",
"det_rotation",
"linear_equiv.coe_det",
"rotation",
"units.coe_one"
] | The determinant of `rotation` (as a linear equiv) is equal to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_eq_smul_circle_integral [complete_space F] {R : ℝ} {c : ℂ} {f : ℂ → F} (hR : 0 < R)
(hf : diff_cont_on_cl ℂ f (ball c R)) :
deriv f c = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z | begin
lift R to ℝ≥0 using hR.le,
refine (hf.has_fpower_series_on_ball hR).has_fpower_series_at.deriv.trans _,
simp only [cauchy_power_series_apply, one_div, zpow_neg, pow_one, smul_smul, zpow_two, mul_inv]
end | lemma | complex.deriv_eq_smul_circle_integral | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"cauchy_power_series_apply",
"complete_space",
"deriv",
"diff_cont_on_cl",
"lift",
"mul_inv",
"one_div",
"pow_one",
"smul_smul",
"zpow_neg",
"zpow_two"
] | If `f` is complex differentiable on an open disc with center `c` and radius `R > 0` and is
continuous on its closure, then `f' c` can be represented as an integral over the corresponding
circle.
TODO: add a version for `w ∈ metric.ball c R`.
TODO: add a version for higher derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_deriv_le_aux [complete_space F] {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
(hf : diff_cont_on_cl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
‖deriv f c‖ ≤ C / R | begin
have : ∀ z ∈ sphere c R, ‖(z - c) ^ (-2 : ℤ) • f z‖ ≤ C / (R * R),
from λ z (hz : abs (z - c) = R), by simpa [-mul_inv_rev, norm_smul, hz, zpow_two,
←div_eq_inv_mul] using (div_le_div_right (mul_pos hR hR)).2 (hC z hz),
calc ‖deriv f c‖ = ‖(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z... | lemma | complex.norm_deriv_le_aux | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"circle_integral.norm_two_pi_I_inv_smul_integral_le_of_norm_le_const",
"complete_space",
"diff_cont_on_cl",
"div_eq_mul_inv",
"div_le_div_right",
"div_self_mul_self'",
"mul_div_left_comm",
"mul_inv_rev",
"norm_smul",
"zpow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
(hd : diff_cont_on_cl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
‖deriv f c‖ ≤ C / R | begin
set e : F →L[ℂ] F̂ := uniform_space.completion.to_complL,
have : has_deriv_at (e ∘ f) (e (deriv f c)) c,
from e.has_fderiv_at.comp_has_deriv_at c
(hd.differentiable_at is_open_ball $ mem_ball_self hR).has_deriv_at,
calc ‖deriv f c‖ = ‖deriv (e ∘ f) c‖ :
by { rw this.deriv, exact (uniform_space... | lemma | complex.norm_deriv_le_of_forall_mem_sphere_norm_le | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"deriv",
"diff_cont_on_cl",
"has_deriv_at",
"uniform_space.completion.norm_coe",
"uniform_space.completion.to_complL"
] | If `f` is complex differentiable on an open disc of radius `R > 0`, is continuous on its
closure, and its values on the boundary circle of this disc are bounded from above by `C`, then the
norm of its derivative at the center is at most `C / R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
liouville_theorem_aux {f : ℂ → F} (hf : differentiable ℂ f)
(hb : bounded (range f)) (z w : ℂ) : f z = f w | begin
suffices : ∀ c, deriv f c = 0, from is_const_of_deriv_eq_zero hf this z w,
clear z w, intro c,
obtain ⟨C, C₀, hC⟩ : ∃ C > (0 : ℝ), ∀ z, ‖f z‖ ≤ C,
{ rcases bounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩,
exact ⟨max C 1, lt_max_iff.2 (or.inr zero_lt_one),
λ z, (hC (f z) (mem_range_self _)).trans (l... | lemma | complex.liouville_theorem_aux | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"deriv",
"differentiable",
"div_div_cancel'",
"div_pos",
"is_const_of_deriv_eq_zero",
"le_of_forall_le_of_dense",
"zero_lt_one"
] | An auxiliary lemma for Liouville's theorem `differentiable.apply_eq_apply_of_bounded`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_eq_apply_of_bounded {f : E → F} (hf : differentiable ℂ f) (hb : bounded (range f))
(z w : E) : f z = f w | begin
set g : ℂ → F := f ∘ (λ t : ℂ, t • (w - z) + z),
suffices : g 0 = g 1, by simpa [g],
apply liouville_theorem_aux,
exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z),
hb.mono (range_comp_subset_range _ _)]
end | lemma | differentiable.apply_eq_apply_of_bounded | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"differentiable"
] | **Liouville's theorem**: a complex differentiable bounded function `f : E → F` is a constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_const_forall_eq_of_bounded {f : E → F} (hf : differentiable ℂ f)
(hb : bounded (range f)) : ∃ c, ∀ z, f z = c | ⟨f 0, λ z, hf.apply_eq_apply_of_bounded hb _ _⟩ | lemma | differentiable.exists_const_forall_eq_of_bounded | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"differentiable"
] | **Liouville's theorem**: a complex differentiable bounded function is a constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_eq_const_of_bounded {f : E → F} (hf : differentiable ℂ f)
(hb : bounded (range f)) : ∃ c, f = const E c | (hf.exists_const_forall_eq_of_bounded hb).imp $ λ c, funext | lemma | differentiable.exists_eq_const_of_bounded | analysis.complex | src/analysis/complex/liouville.lean | [
"analysis.complex.cauchy_integral",
"analysis.calculus.fderiv_analytic",
"analysis.normed_space.completion"
] | [
"differentiable"
] | **Liouville's theorem**: a complex differentiable bounded function is a constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cderiv (r : ℝ) (f : ℂ → E) (z : ℂ) : E | (2 * π * I : ℂ)⁻¹ • ∮ w in C(z, r), ((w - z) ^ 2)⁻¹ • f w | def | complex.cderiv | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [] | A circle integral which coincides with `deriv f z` whenever one can apply the Cauchy formula for
the derivative. It is useful in the proof that locally uniform limits of holomorphic functions are
holomorphic, because it depends continuously on `f` for the uniform topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cderiv_eq_deriv (hU : is_open U) (hf : differentiable_on ℂ f U) (hr : 0 < r)
(hzr : closed_ball z r ⊆ U) :
cderiv r f z = deriv f z | two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable hU hzr hf (mem_ball_self hr) | lemma | complex.cderiv_eq_deriv | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"deriv",
"differentiable_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
‖cderiv r f z‖ ≤ M / r | begin
have hM : 0 ≤ M,
{ obtain ⟨w, hw⟩ : (sphere z r).nonempty := normed_space.sphere_nonempty.mpr hr.le,
exact (norm_nonneg _).trans (hf w hw) },
have h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2,
{ intros w hw,
simp only [mem_sphere_iff_norm, norm_eq_abs] at hw,
simp only [norm_smul... | lemma | complex.norm_cderiv_le | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"circle_integral.norm_integral_le_of_norm_le_const",
"complex.abs_pow",
"div_le_div",
"inv_mul_eq_div",
"le_rfl",
"map_inv₀",
"mul_le_mul",
"norm_smul",
"ring",
"sq_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cderiv_sub (hr : 0 < r) (hf : continuous_on f (sphere z r))
(hg : continuous_on g (sphere z r)) :
cderiv r (f - g) z = cderiv r f z - cderiv r g z | begin
have h1 : continuous_on (λ (w : ℂ), ((w - z) ^ 2)⁻¹) (sphere z r),
{ refine ((continuous_id'.sub continuous_const).pow 2).continuous_on.inv₀ (λ w hw h, hr.ne _),
rwa [mem_sphere_iff_norm, sq_eq_zero_iff.mp h, norm_zero] at hw },
simp_rw [cderiv, ← smul_sub],
congr' 1,
simpa only [pi.sub_apply, smul_... | lemma | complex.cderiv_sub | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"circle_integrable",
"circle_integral.integral_sub",
"continuous_const",
"continuous_on",
"continuous_on.inv₀",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : continuous_on f (sphere z r)) :
‖cderiv r f z‖ < M / r | begin
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L,
{ have e1 : (sphere z r).nonempty := normed_space.sphere_nonempty.mpr hr.le,
have e2 : continuous_on (λ w, ‖f w‖) (sphere z r),
from continuous_norm.comp_continuous_on hf,
obtain ⟨x, hx, hx'⟩ := (is_compact_sphere z r).exists_forall_ge... | lemma | complex.norm_cderiv_lt | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"continuous_on",
"div_lt_div_right",
"is_compact_sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cderiv_sub_lt (hr : 0 < r) (hfg : ∀ w ∈ sphere z r, ‖f w - g w‖ < M)
(hf : continuous_on f (sphere z r)) (hg : continuous_on g (sphere z r)) :
‖cderiv r f z - cderiv r g z‖ < M / r | cderiv_sub hr hf hg ▸ norm_cderiv_lt hr hfg (hf.sub hg) | lemma | complex.norm_cderiv_sub_lt | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.cderiv (hF : tendsto_uniformly_on F f φ (cthickening δ K)) (hδ : 0 < δ)
(hFn : ∀ᶠ n in φ, continuous_on (F n) (cthickening δ K)) :
tendsto_uniformly_on (cderiv δ ∘ F) (cderiv δ f) φ K | begin
by_cases φ = ⊥,
{ simp only [h, tendsto_uniformly_on, eventually_bot, implies_true_iff]},
haveI : φ.ne_bot := ne_bot_iff.2 h,
have e1 : continuous_on f (cthickening δ K) := tendsto_uniformly_on.continuous_on hF hFn,
rw [tendsto_uniformly_on_iff] at hF ⊢,
rintro ε hε,
filter_upwards [hF (ε * δ) (mul_... | lemma | complex.tendsto_uniformly_on.cderiv | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"continuous_on",
"mul_div_cancel",
"tendsto_uniformly_on",
"tendsto_uniformly_on.continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_deriv_of_cthickening_subset (hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ n in φ, differentiable_on ℂ (F n) U) {δ : ℝ} (hδ: 0 < δ) (hK : is_compact K)
(hU : is_open U) (hKU : cthickening δ K ⊆ U) :
tendsto_uniformly_on (deriv ∘ F) (cderiv δ f) φ K | begin
have h1 : ∀ᶠ n in φ, continuous_on (F n) (cthickening δ K),
by filter_upwards [hF] with n h using h.continuous_on.mono hKU,
have h2 : is_compact (cthickening δ K),
from is_compact_of_is_closed_bounded is_closed_cthickening hK.bounded.cthickening,
have h3 : tendsto_uniformly_on F f φ (cthickening δ K... | lemma | complex.tendsto_uniformly_on_deriv_of_cthickening_subset | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"continuous_on",
"deriv",
"differentiable_on",
"is_compact",
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_iff_forall_is_compact",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_cthickening_tendsto_uniformly_on (hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ n in φ, differentiable_on ℂ (F n) U) (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :
∃ δ > 0, cthickening δ K ⊆ U ∧ tendsto_uniformly_on (deriv ∘ F) (cderiv δ f) φ K | begin
obtain ⟨δ, hδ, hKδ⟩ := hK.exists_cthickening_subset_open hU hKU,
exact ⟨δ, hδ, hKδ, tendsto_uniformly_on_deriv_of_cthickening_subset hf hF hδ hK hU hKδ⟩
end | lemma | complex.exists_cthickening_tendsto_uniformly_on | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"deriv",
"differentiable_on",
"is_compact",
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.tendsto_locally_uniformly_on.differentiable_on [φ.ne_bot]
(hf : tendsto_locally_uniformly_on F f φ U) (hF : ∀ᶠ n in φ, differentiable_on ℂ (F n) U)
(hU : is_open U) :
differentiable_on ℂ f U | begin
rintro x hx,
obtain ⟨K, ⟨hKx, hK⟩, hKU⟩ := (compact_basis_nhds x).mem_iff.mp (hU.mem_nhds hx),
obtain ⟨δ, hδ, -, h1⟩ := exists_cthickening_tendsto_uniformly_on hf hF hK hU hKU,
have h2 : interior K ⊆ U := interior_subset.trans hKU,
have h3 : ∀ᶠ n in φ, differentiable_on ℂ (F n) (interior K),
filter_... | theorem | tendsto_locally_uniformly_on.differentiable_on | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"compact_basis_nhds",
"deriv",
"differentiable_at.differentiable_within_at",
"differentiable_on",
"differentiable_within_at",
"has_deriv_at",
"has_deriv_at_of_tendsto_locally_uniformly_on'",
"interior",
"interior_subset",
"is_open",
"is_open_interior",
"tendsto_locally_uniformly_on"
] | A locally uniform limit of holomorphic functions on an open domain of the complex plane is
holomorphic (the derivatives converge locally uniformly to that of the limit, which is proved
as `tendsto_locally_uniformly_on.deriv`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.tendsto_locally_uniformly_on.deriv (hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ n in φ, differentiable_on ℂ (F n) U) (hU : is_open U) :
tendsto_locally_uniformly_on (deriv ∘ F) (deriv f) φ U | begin
rw [tendsto_locally_uniformly_on_iff_forall_is_compact hU],
by_cases φ = ⊥,
{ simp only [h, tendsto_uniformly_on, eventually_bot, implies_true_iff] },
haveI : φ.ne_bot := ne_bot_iff.2 h,
rintro K hKU hK,
obtain ⟨δ, hδ, hK4, h⟩ := exists_cthickening_tendsto_uniformly_on hf hF hK hU hKU,
refine h.cong... | lemma | tendsto_locally_uniformly_on.deriv | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"deriv",
"differentiable_on",
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_iff_forall_is_compact",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_tsum_of_summable_norm {u : ι → ℝ}
(hu : summable u) (hf : ∀ (i : ι), differentiable_on ℂ (F i) U) (hU : is_open U)
(hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) :
differentiable_on ℂ (λ w : ℂ, ∑' (i : ι), F i w) U | begin
classical,
have hc := (tendsto_uniformly_on_tsum hu hF_le).tendsto_locally_uniformly_on,
refine hc.differentiable_on (eventually_of_forall $ λ s, _) hU,
exact differentiable_on.sum (λ i hi, hf i),
end | lemma | complex.differentiable_on_tsum_of_summable_norm | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"differentiable_on",
"differentiable_on.sum",
"is_open",
"summable",
"tendsto_locally_uniformly_on",
"tendsto_uniformly_on_tsum"
] | If the terms in the sum `∑' (i : ι), F i` are uniformly bounded on `U` by a
summable function, and each term in the sum is differentiable on `U`, then so is the sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_deriv_of_summable_norm {u : ι → ℝ}
(hu : summable u) (hf : ∀ (i : ι), differentiable_on ℂ (F i) U) (hU : is_open U)
(hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) (hz : z ∈ U) :
has_sum (λ (i : ι), deriv (F i) z) (deriv (λ w : ℂ, ∑' (i : ι), F i w) z) | begin
rw has_sum,
have hc := (tendsto_uniformly_on_tsum hu hF_le).tendsto_locally_uniformly_on,
convert (hc.deriv (eventually_of_forall $ λ s, differentiable_on.sum
(λ i hi, hf i)) hU).tendsto_at hz using 1,
ext1 s,
exact (deriv_sum (λ i hi, (hf i).differentiable_at (hU.mem_nhds hz))).symm,
end | lemma | complex.has_sum_deriv_of_summable_norm | analysis.complex | src/analysis/complex/locally_uniform_limit.lean | [
"analysis.complex.removable_singularity",
"analysis.calculus.series"
] | [
"deriv",
"deriv_sum",
"differentiable_at",
"differentiable_on",
"differentiable_on.sum",
"has_sum",
"is_open",
"summable",
"tendsto_locally_uniformly_on",
"tendsto_uniformly_on_tsum"
] | If the terms in the sum `∑' (i : ι), F i` are uniformly bounded on `U` by a
summable function, then the sum of `deriv F i` at a point in `U` is the derivative of the
sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diff_cont_on_cl.ball_subset_image_closed_ball (h : diff_cont_on_cl ℂ f (ball z₀ r))
(hr : 0 < r) (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
ball (f z₀) (ε / 2) ⊆ f '' closed_ball z₀ r | begin
/- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value at `z₀`
so it is not constant on the disk, which implies that its infimum is equal to `0` and hence that
`v` is in the r... | lemma | diff_cont_on_cl.ball_subset_image_closed_ball | analysis.complex | src/analysis/complex/open_mapping.lean | [
"analysis.analytic.isolated_zeros",
"analysis.complex.cauchy_integral",
"analysis.complex.abs_max"
] | [
"analytic_on",
"analytic_on_const",
"closure_ball",
"continuous_on",
"convex_ball",
"diff_cont_on_cl",
"dist_comm",
"is_local_min",
"is_preconnected",
"nhds_within_le_nhds"
] | If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_at.eventually_constant_or_nhds_le_map_nhds_aux (hf : analytic_at ℂ f z₀) :
(∀ᶠ z in 𝓝 z₀, f z = f z₀) ∨ (𝓝 (f z₀) ≤ map f (𝓝 z₀)) | begin
/- The function `f` is analytic in a neighborhood of `z₀`; by the isolated zeros principle, if `f`
is not constant in a neighborhood of `z₀`, then it is nonzero, and therefore bounded below, on
every small enough circle around `z₀` and then `diff_cont_on_cl.ball_subset_image_closed_ball`
provides an expli... | lemma | analytic_at.eventually_constant_or_nhds_le_map_nhds_aux | analysis.complex | src/analysis/complex/open_mapping.lean | [
"analysis.analytic.isolated_zeros",
"analysis.complex.cauchy_integral",
"analysis.complex.abs_max"
] | [
"analytic_at",
"analytic_at_const",
"analytic_on",
"closure_ball",
"continuous_on",
"diff_cont_on_cl",
"half_pos",
"inf_le_left",
"inf_le_right",
"is_compact_sphere",
"is_open_analytic_at"
] | A function `f : ℂ → ℂ` which is analytic at a point `z₀` is either constant in a neighborhood
of `z₀`, or behaves locally like an open function (in the sense that the image of every neighborhood
of `z₀` is a neighborhood of `f z₀`, as in `is_open_map_iff_nhds_le`). For a function `f : E → ℂ`
the same result holds, see ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_at.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : analytic_at ℂ g z₀) :
(∀ᶠ z in 𝓝 z₀, g z = g z₀) ∨ (𝓝 (g z₀) ≤ map g (𝓝 z₀)) | begin
/- The idea of the proof is to use the one-dimensional version applied to the restriction of `g`
to lines going through `z₀` (indexed by `sphere (0 : E) 1`). If the restriction is eventually
constant along each of these lines, then the identity theorem implies that `g` is constant on any
ball centered at ... | lemma | analytic_at.eventually_constant_or_nhds_le_map_nhds | analysis.complex | src/analysis/complex/open_mapping.lean | [
"analysis.analytic.isolated_zeros",
"analysis.complex.cauchy_integral",
"analysis.complex.abs_max"
] | [
"abs_norm",
"analytic_at",
"analytic_at.comp",
"analytic_on",
"analytic_on_const",
"continuous",
"continuous_const",
"continuous_linear_map.id",
"continuous_linear_map.smul_right",
"convex_ball",
"eq_or_ne",
"is_open_analytic_at",
"is_preconnected",
"mul_inv_cancel",
"norm_eq_zero",
"n... | The *open mapping theorem* for holomorphic functions, local version: is a function `g : E → ℂ`
is analytic at a point `z₀`, then either it is constant in a neighborhood of `z₀`, or it maps every
neighborhood of `z₀` to a neighborhood of `z₀`. For the particular case of a holomorphic function on
`ℂ`, see `analytic_at.ev... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on.is_constant_or_is_open (hg : analytic_on ℂ g U) (hU : is_preconnected U) :
(∃ w, ∀ z ∈ U, g z = w) ∨ (∀ s ⊆ U, is_open s → is_open (g '' s)) | begin
by_cases ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀,
{ obtain ⟨z₀, hz₀, h⟩ := h,
exact or.inl ⟨g z₀, hg.eq_on_of_preconnected_of_eventually_eq analytic_on_const hU hz₀ h⟩ },
{ push_neg at h,
refine or.inr (λ s hs1 hs2, is_open_iff_mem_nhds.mpr _),
rintro z ⟨w, hw1, rfl⟩,
exact (hg w (hs1 hw1)).even... | theorem | analytic_on.is_constant_or_is_open | analysis.complex | src/analysis/complex/open_mapping.lean | [
"analysis.analytic.isolated_zeros",
"analysis.complex.cauchy_integral",
"analysis.complex.abs_max"
] | [
"analytic_on",
"analytic_on_const",
"is_open",
"is_preconnected"
] | The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
sense that it maps any open set contained in `U` to an open set in `ℂ`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_conj_lie : (conj_lie.to_linear_equiv : ℂ →ₗ[ℝ] ℂ).det = -1 | det_conj_ae | lemma | complex.det_conj_lie | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [] | The determinant of `conj_lie`, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_det_conj_lie : conj_lie.to_linear_equiv.det = -1 | linear_equiv_det_conj_ae | lemma | complex.linear_equiv_det_conj_lie | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [] | The determinant of `conj_lie`, as a linear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_clm_norm : ‖re_clm‖ = 1 | le_antisymm (linear_map.mk_continuous_norm_le _ zero_le_one _) $
calc 1 = ‖re_clm 1‖ : by simp
... ≤ ‖re_clm‖ : unit_le_op_norm _ _ (by simp) | lemma | complex.re_clm_norm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"linear_map.mk_continuous_norm_le",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_clm_nnnorm : ‖re_clm‖₊ = 1 | subtype.ext re_clm_norm | lemma | complex.re_clm_nnnorm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_clm_norm : ‖im_clm‖ = 1 | le_antisymm (linear_map.mk_continuous_norm_le _ zero_le_one _) $
calc 1 = ‖im_clm I‖ : by simp
... ≤ ‖im_clm‖ : unit_le_op_norm _ _ (by simp) | lemma | complex.im_clm_norm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"linear_map.mk_continuous_norm_le",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_clm_nnnorm : ‖im_clm‖₊ = 1 | subtype.ext im_clm_norm | lemma | complex.im_clm_nnnorm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_cle_norm : ‖(conj_cle : ℂ →L[ℝ] ℂ)‖ = 1 | conj_lie.to_linear_isometry.norm_to_continuous_linear_map | lemma | complex.conj_cle_norm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_cle_nnorm : ‖(conj_cle : ℂ →L[ℝ] ℂ)‖₊ = 1 | subtype.ext conj_cle_norm | lemma | complex.conj_cle_nnorm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_clm_norm : ‖of_real_clm‖ = 1 | of_real_li.norm_to_continuous_linear_map | lemma | complex.of_real_clm_norm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_clm_nnnorm : ‖of_real_clm‖₊ = 1 | subtype.ext $ of_real_clm_norm | lemma | complex.of_real_clm_nnnorm | analysis.complex | src/analysis/complex/operator_norm.lean | [
"analysis.complex.basic",
"analysis.normed_space.operator_norm",
"data.complex.determinant"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_sub_exp_exp {a : ℝ} {f g : ℂ → E} {l : filter ℂ} {u : ℂ → ℝ}
(hBf : ∃ (c < a) B, f =O[l] (λ z, expR (B * expR (c * |u z|))))
(hBg : ∃ (c < a) B, g =O[l] (λ z, expR (B * expR (c * |u z|)))) :
∃ (c < a) B, (f - g) =O[l] (λ z, expR (B * expR (c * |u z|))) | begin
have : ∀ {c₁ c₂ B₁ B₂}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ z,
‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖,
{ intros c₁ c₂ B₁ B₂ hc hB₀ hB z,
rw [real.norm_eq_abs, real.norm_eq_abs, real.abs_exp, real.abs_exp, real.exp_le_exp],
exact mul_le_mul hB (real.exp_le_exp.2 $ mul_le_mul_o... | lemma | phragmen_lindelof.is_O_sub_exp_exp | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"abs_nonneg",
"filter",
"mul_le_mul",
"mul_le_mul_of_nonneg_right",
"real.abs_exp",
"real.exp_le_exp",
"real.exp_pos",
"real.norm_eq_abs"
] | An auxiliary lemma that combines two double exponential estimates into a similar estimate
on the difference of the functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_sub_exp_rpow {a : ℝ} {f g : ℂ → E} {l : filter ℂ}
(hBf : ∃ (c < a) B, f =O[comap complex.abs at_top ⊓ l] (λ z, expR (B * (abs z) ^ c)))
(hBg : ∃ (c < a) B, g =O[comap complex.abs at_top ⊓ l] (λ z, expR (B * (abs z) ^ c))) :
∃ (c < a) B, (f - g) =O[comap complex.abs at_top ⊓ l] (λ z, expR (B * (abs z) ^ c)) | begin
have : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ →
(λ z : ℂ, expR (B₁ * (abs z) ^ c₁)) =O[comap complex.abs at_top ⊓ l]
(λ z, expR (B₂ * (abs z) ^ c₂)),
{ have : ∀ᶠ z : ℂ in comap complex.abs at_top ⊓ l, 1 ≤ abs z,
from ((eventually_ge_at_top 1).comap _).filter_mono inf_le_left,
refi... | lemma | phragmen_lindelof.is_O_sub_exp_rpow | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"filter",
"inf_le_left",
"mul_le_mul",
"one_mul",
"real.abs_exp",
"real.exp_le_exp",
"real.norm_eq_abs",
"real.rpow_le_rpow_of_exponent_le",
"real.rpow_nonneg_of_nonneg"
] | An auxiliary lemma that combines two “exponential of a power” estimates into a similar estimate
on the difference of the functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
horizontal_strip (hfd : diff_cont_on_cl ℂ f (im ⁻¹' Ioo a b))
(hB : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ re) at_top ⊓ 𝓟 (im ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.re|))))
(hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C)
(hza : a ≤ im z) (hzb : im z ≤ b) :
‖f z‖ ≤ ... | begin
-- If `im z = a` or `im z = b`, then we apply `hle_a` or `hle_b`, otherwise `im z ∈ Ioo a b`.
rw le_iff_eq_or_lt at hza hzb,
cases hza with hza hza, { exact hle_a _ hza.symm },
cases hzb with hzb hzb, { exact hle_b _ hzb },
-- WLOG, `0 < C`.
suffices : ∀ C' : ℝ, 0 < C' → (∀ w : ℂ, im w = a → ‖f w‖ ≤ C... | lemma | phragmen_lindelof.horizontal_strip | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"abs_eq",
"abs_lt",
"abs_of_pos",
"closure_Ioo",
"continuous_const",
"diff_cont_on_cl",
"differentiable",
"div_eq_inv_mul",
"div_mul_eq_div_div",
"div_pos",
"exists_between",
"exp",
"frontier_Ioo",
"frontier_subset_closure",
"ge_mem_nhds",
"le_iff_eq_or_lt",
"le_of_forall_le_of_dense... | **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < im z < b}`.
Let `f : ℂ → E` be a function such that
* `f` is differentiable on `U` and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * exp(c * |re z|))` on `U` for some `c < π / (b - a)`;
* `‖f z‖` is bounded from above by a con... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_horizontal_strip (hd : diff_cont_on_cl ℂ f (im ⁻¹' Ioo a b))
(hB : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ re) at_top ⊓ 𝓟 (im ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.re|))))
(ha : ∀ z : ℂ, z.im = a → f z = 0) (hb : ∀ z : ℂ, z.im = b → f z = 0) :
eq_on f 0 (im ⁻¹' Icc a b) | λ z hz, norm_le_zero_iff.1 $ horizontal_strip hd hB
(λ z hz, (ha z hz).symm ▸ norm_zero.le) (λ z hz, (hb z hz).symm ▸ norm_zero.le) hz.1 hz.2 | lemma | phragmen_lindelof.eq_zero_on_horizontal_strip | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < im z < b}`.
Let `f : ℂ → E` be a function such that
* `f` is differentiable on `U` and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * exp(c * |re z|))` on `U` for some `c < π / (b - a)`;
* `f z = 0` on the boundary of `U`.
The... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_horizontal_strip {g : ℂ → E} (hdf : diff_cont_on_cl ℂ f (im ⁻¹' Ioo a b))
(hBf : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ re) at_top ⊓ 𝓟 (im ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.re|))))
(hdg : diff_cont_on_cl ℂ g (im ⁻¹' Ioo a b))
(hBg : ∃ (c < π / (b - a)) B, g =O[comap (has_abs.abs ∘ re... | λ z hz, sub_eq_zero.1 (eq_zero_on_horizontal_strip (hdf.sub hdg) (is_O_sub_exp_exp hBf hBg)
(λ w hw, sub_eq_zero.2 (ha w hw)) (λ w hw, sub_eq_zero.2 (hb w hw)) hz) | lemma | phragmen_lindelof.eq_on_horizontal_strip | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < im z < b}`.
Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable on `U` and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * exp(c * |re z|))` on `U` for some
`c < π / (b - a)`;
* `f z = g z` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.