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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vertical_strip (hfd : diff_cont_on_cl ℂ f (re ⁻¹' Ioo a b))
(hB : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ im) at_top ⊓ 𝓟 (re ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.im|))))
(hle_a : ∀ z : ℂ, re z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, re z = b → ‖f z‖ ≤ C)
(hza : a ≤ re z) (hzb : re z ≤ b) :
‖f z‖ ≤ C | begin
suffices : ‖(λ z, f (z * (-I))) (z * I)‖ ≤ C, by simpa [mul_assoc] using this,
have H : maps_to (λ z, z * (-I)) (im ⁻¹' Ioo a b) (re ⁻¹' Ioo a b),
{ intros z hz, simpa using hz },
refine horizontal_strip (hfd.comp (differentiable_id.mul_const _).diff_cont_on_cl H)
_ (λ z hz, hle_a _ _) (λ z hz, hle_b ... | lemma | phragmen_lindelof.vertical_strip | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"diff_cont_on_cl",
"mul_assoc"
] | **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < re 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 * |im 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_vertical_strip (hd : diff_cont_on_cl ℂ f (re ⁻¹' Ioo a b))
(hB : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ im) at_top ⊓ 𝓟 (re ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.im|))))
(ha : ∀ z : ℂ, re z = a → f z = 0) (hb : ∀ z : ℂ, re z = b → f z = 0) :
eq_on f 0 (re ⁻¹' Icc a b) | λ z hz, norm_le_zero_iff.1 $ vertical_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_vertical_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 < re 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 * |im 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_vertical_strip {g : ℂ → E} (hdf : diff_cont_on_cl ℂ f (re ⁻¹' Ioo a b))
(hBf : ∃ (c < π / (b - a)) B, f =O[comap (has_abs.abs ∘ im) at_top ⊓ 𝓟 (re ⁻¹' Ioo a b)]
(λ z, expR (B * expR (c * |z.im|))))
(hdg : diff_cont_on_cl ℂ g (re ⁻¹' Ioo a b))
(hBg : ∃ (c < π / (b - a)) B, g =O[comap (has_abs.abs ∘ im) ... | λ z hz, sub_eq_zero.1 (eq_zero_on_vertical_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_vertical_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 < re 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 * |im z|))` on `U` for some
`c < π / (b - a)`;
* `f z = g z` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quadrant_I (hd : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Ioi 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C)
(hz_re : 0 ≤ z.re) (hz_im : 0 ≤ z.im) :
‖f z‖ ≤ C | begin
-- The case `z = 0` is trivial.
rcases eq_or_ne z 0 with rfl|hzne, { exact hre 0 le_rfl },
-- Otherwise, `z = e ^ ζ` for some `ζ : ℂ`, `0 < Im ζ < π / 2`.
obtain ⟨ζ, hζ, rfl⟩ : ∃ ζ : ℂ, ζ.im ∈ Icc 0 (π / 2) ∧ exp ζ = z,
{ refine ⟨log z, _, exp_log hzne⟩,
rw log_im,
exact ⟨arg_nonneg_iff.2 hz_im,... | lemma | phragmen_lindelof.quadrant_I | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"continuous_within_at",
"diff_cont_on_cl",
"div_div_cancel'",
"div_pos",
"eq_or_ne",
"exp",
"inf_sup_right",
"le_rfl",
"lift",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"one_mul",
"real.cos_pi_div_two",
"real.cos_pos_of_mem_Ioo",
"real.exp_le_exp",
"real.exp_mul",
"r... | **Phragmen-Lindelöf principle** in the first quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open first quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open first quadrant
for some `c < 2`;
* `‖f z‖` is bounded from abov... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_quadrant_I (hd : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Ioi 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, 0 ≤ x → f x = 0) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = 0) :
eq_on f 0 {z | 0 ≤ z.re ∧ 0 ≤ z.im} | λ z hz, norm_le_zero_iff.1 $ quadrant_I hd hB (λ x hx, norm_le_zero_iff.2 $ hre x hx)
(λ x hx, norm_le_zero_iff.2 $ him x hx) hz.1 hz.2 | lemma | phragmen_lindelof.eq_zero_on_quadrant_I | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the first quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open first quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open first quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equal t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_quadrant_I (hdf : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Ioi 0))
(hBf : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hdg : diff_cont_on_cl ℂ g (Ioi 0 ×ℂ Ioi 0))
(hBg : ∃ (c < (2 : ℝ)) B, g =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)]
(λ z, expR... | λ z hz, sub_eq_zero.1 $ eq_zero_on_quadrant_I (hdf.sub hdg) (is_O_sub_exp_rpow hBf hBg)
(λ x hx, sub_eq_zero.2 $ hre x hx) (λ x hx, sub_eq_zero.2 $ him x hx) hz | lemma | phragmen_lindelof.eq_on_quadrant_I | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the first quadrant. Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable in the open first quadrant and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * (abs z) ^ c)` on the open first
quadrant for some `A`, `B`, and `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quadrant_II (hd : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Ioi 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C)
(hz_re : z.re ≤ 0) (hz_im : 0 ≤ z.im) :
‖f z‖ ≤ C | begin
obtain ⟨z, rfl⟩ : ∃ z', z' * I = z, from ⟨z / I, div_mul_cancel _ I_ne_zero⟩,
simp only [mul_I_re, mul_I_im, neg_nonpos] at hz_re hz_im,
change ‖(f ∘ (* I)) z‖ ≤ C,
have H : maps_to (* I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0),
{ intros w hw,
simpa only [mem_re_prod_im, mul_I_re, mul_I_im, neg_lt_zero, m... | lemma | phragmen_lindelof.quadrant_II | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"complex.abs",
"diff_cont_on_cl",
"div_mul_cancel",
"map_mul",
"mul_assoc",
"mul_neg_one",
"mul_one"
] | **Phragmen-Lindelöf principle** in the second quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open second quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open second quadrant
for some `c < 2`;
* `‖f z‖` is bounded from a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_quadrant_II (hd : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Ioi 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, x ≤ 0 → f x = 0) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = 0) :
eq_on f 0 {z | z.re ≤ 0 ∧ 0 ≤ z.im} | λ z hz, norm_le_zero_iff.1 $ quadrant_II hd hB (λ x hx, norm_le_zero_iff.2 $ hre x hx)
(λ x hx, norm_le_zero_iff.2 $ him x hx) hz.1 hz.2 | lemma | phragmen_lindelof.eq_zero_on_quadrant_II | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the second quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open second quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open second quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equa... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_quadrant_II (hdf : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Ioi 0))
(hBf : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)]
(λ z, expR (B * (abs z) ^ c)))
(hdg : diff_cont_on_cl ℂ g (Iio 0 ×ℂ Ioi 0))
(hBg : ∃ (c < (2 : ℝ)) B, g =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)]
(λ z, exp... | λ z hz, sub_eq_zero.1 $ eq_zero_on_quadrant_II (hdf.sub hdg) (is_O_sub_exp_rpow hBf hBg)
(λ x hx, sub_eq_zero.2 $ hre x hx) (λ x hx, sub_eq_zero.2 $ him x hx) hz | lemma | phragmen_lindelof.eq_on_quadrant_II | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the second quadrant. Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable in the open second quadrant and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * (abs z) ^ c)` on the open second
quadrant for some `A`, `B`, an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quadrant_III (hd : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C)
(hz_re : z.re ≤ 0) (hz_im : z.im ≤ 0) :
‖f z‖ ≤ C | begin
obtain ⟨z, rfl⟩ : ∃ z', -z' = z, from ⟨-z, neg_neg z⟩,
simp only [neg_re, neg_im, neg_nonpos] at hz_re hz_im,
change ‖(f ∘ has_neg.neg) z‖ ≤ C,
have H : maps_to has_neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0),
{ intros w hw,
simpa only [mem_re_prod_im, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw },... | lemma | phragmen_lindelof.quadrant_III | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"complex.abs",
"diff_cont_on_cl",
"neg_mul"
] | **Phragmen-Lindelöf principle** in the third quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open third quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp (B * (abs z) ^ c)` on the open third quadrant
for some `c < 2`;
* `‖f z‖` is bounded from abo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_quadrant_III (hd : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, x ≤ 0 → f x = 0) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = 0) :
eq_on f 0 {z | z.re ≤ 0 ∧ z.im ≤ 0} | λ z hz, norm_le_zero_iff.1 $ quadrant_III hd hB (λ x hx, norm_le_zero_iff.2 $ hre x hx)
(λ x hx, norm_le_zero_iff.2 $ him x hx) hz.1 hz.2 | lemma | phragmen_lindelof.eq_zero_on_quadrant_III | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the third quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open third quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open third quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equal t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_quadrant_III (hdf : diff_cont_on_cl ℂ f (Iio 0 ×ℂ Iio 0))
(hBf : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hdg : diff_cont_on_cl ℂ g (Iio 0 ×ℂ Iio 0))
(hBg : ∃ (c < (2 : ℝ)) B, g =O[comap complex.abs at_top ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)]
(λ z, ex... | λ z hz, sub_eq_zero.1 $ eq_zero_on_quadrant_III (hdf.sub hdg) (is_O_sub_exp_rpow hBf hBg)
(λ x hx, sub_eq_zero.2 $ hre x hx) (λ x hx, sub_eq_zero.2 $ him x hx) hz | lemma | phragmen_lindelof.eq_on_quadrant_III | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the third quadrant. Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable in the open third quadrant and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * (abs z) ^ c)` on the open third
quadrant for some `A`, `B`, and `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quadrant_IV (hd : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Iio 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C)
(hz_re : 0 ≤ z.re) (hz_im : z.im ≤ 0) :
‖f z‖ ≤ C | begin
obtain ⟨z, rfl⟩ : ∃ z', -z' = z, from ⟨-z, neg_neg z⟩,
simp only [neg_re, neg_im, neg_nonpos, neg_nonneg] at hz_re hz_im,
change ‖(f ∘ has_neg.neg) z‖ ≤ C,
have H : maps_to has_neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ Iio 0),
{ intros w hw,
simpa only [mem_re_prod_im, neg_re, neg_im, neg_lt_zero, neg_pos,... | lemma | phragmen_lindelof.quadrant_IV | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"complex.abs",
"diff_cont_on_cl",
"neg_mul"
] | **Phragmen-Lindelöf principle** in the fourth quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open fourth quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open fourth quadrant
for some `c < 2`;
* `‖f z‖` is bounded from a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_quadrant_IV (hd : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Iio 0))
(hB : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hre : ∀ x : ℝ, 0 ≤ x → f x = 0) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = 0) :
eq_on f 0 {z | 0 ≤ z.re ∧ z.im ≤ 0} | λ z hz, norm_le_zero_iff.1 $ quadrant_IV hd hB (λ x hx, norm_le_zero_iff.2 $ hre x hx)
(λ x hx, norm_le_zero_iff.2 $ him x hx) hz.1 hz.2 | lemma | phragmen_lindelof.eq_zero_on_quadrant_IV | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the fourth quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open fourth quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open fourth quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equa... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_quadrant_IV (hdf : diff_cont_on_cl ℂ f (Ioi 0 ×ℂ Iio 0))
(hBf : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)]
(λ z, expR (B * (abs z) ^ c)))
(hdg : diff_cont_on_cl ℂ g (Ioi 0 ×ℂ Iio 0))
(hBg : ∃ (c < (2 : ℝ)) B, g =O[comap complex.abs at_top ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)]
(λ z, exp... | λ z hz, sub_eq_zero.1 $ eq_zero_on_quadrant_IV (hdf.sub hdg) (is_O_sub_exp_rpow hBf hBg)
(λ x hx, sub_eq_zero.2 $ hre x hx) (λ x hx, sub_eq_zero.2 $ him x hx) hz | lemma | phragmen_lindelof.eq_on_quadrant_IV | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl"
] | **Phragmen-Lindelöf principle** in the fourth quadrant. Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable in the open fourth quadrant and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * (abs z) ^ c)` on the open fourth
quadrant for some `A`, `B`, an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_half_plane_of_tendsto_zero_on_real (hd : diff_cont_on_cl ℂ f {z | 0 < z.re})
(hexp : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 {z | 0 < z.re}]
(λ z, expR (B * (abs z) ^ c)))
(hre : tendsto (λ x : ℝ, f x) at_top (𝓝 0)) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C) (hz : 0 ≤ z.re) :
‖f z‖ ≤ C | begin
/- We are going to apply the Phragmen-Lindelöf principle in the first and fourth quadrants.
The lemmas immediately imply that for any upper estimate `C'` on `‖f x‖`, `x : ℝ`, `0 ≤ x`,
the number `max C C'` is an upper estimate on `f` in the whole right half-plane. -/
revert z,
have hle : ∀ C', (∀ x : ℝ,... | lemma | phragmen_lindelof.right_half_plane_of_tendsto_zero_on_real | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"bot_sup_eq",
"complex.abs",
"complex.abs_of_nonneg",
"continuous_on",
"diff_cont_on_cl",
"exists_prop",
"ge_mem_nhds",
"inf_le_inf_left",
"inf_le_left",
"inf_sup_right",
"is_closed_Ici",
"is_max_on",
"le_abs_self",
"le_rfl",
"real.cocompact_eq"
] | **Phragmen-Lindelöf principle** in the right half-plane. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open right half-plane and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open right half-plane
for some `c < 2`;
* `‖f z‖` is bounded fro... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_half_plane_of_bounded_on_real (hd : diff_cont_on_cl ℂ f {z | 0 < z.re})
(hexp : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 {z | 0 < z.re}]
(λ z, expR (B * (abs z) ^ c)))
(hre : is_bounded_under (≤) at_top (λ x : ℝ, ‖f x‖))
(him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C) (hz : 0 ≤ z.re) :
‖f z‖ ≤ C | begin
-- For each `ε < 0`, the function `λ z, exp (ε * z) • f z` satisfies assumptions of
-- `right_half_plane_of_tendsto_zero_on_real`, hence `‖exp (ε * z) • f z‖ ≤ C` for all `ε < 0`.
-- Taking the limit as `ε → 0`, we obtain the required inequality.
suffices : ∀ᶠ ε : ℝ in 𝓝[<] 0, ‖exp (ε * z) • f z‖ ≤ C,
... | lemma | phragmen_lindelof.right_half_plane_of_bounded_on_real | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"Exists₃.imp",
"complex.abs",
"continuous_const",
"diff_cont_on_cl",
"exp",
"le_of_tendsto",
"mul_le_of_le_one_left",
"mul_nonpos_of_nonpos_of_nonneg",
"mul_zero",
"nhds_within_le_nhds",
"norm_smul",
"one_mul",
"real.exp_zero",
"self_mem_nhds_within"
] | **Phragmen-Lindelöf principle** in the right half-plane. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open right half-plane and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open right half-plane
for some `c < 2`;
* `‖f z‖` is bounded fro... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_on_right_half_plane_of_superexponential_decay
(hd : diff_cont_on_cl ℂ f {z | 0 < z.re})
(hexp : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 {z | 0 < z.re}]
(λ z, expR (B * (abs z) ^ c)))
(hre : superpolynomial_decay at_top expR (λ x, ‖f x‖))
(him : ∃ C, ∀ x : ℝ, ‖f (x * I)‖ ≤ C) :
eq_on ... | begin
rcases him with ⟨C, hC⟩,
-- Due to continuity, it suffices to prove the equality on the open right half-plane.
suffices : ∀ z : ℂ, 0 < z.re → f z = 0,
{ simpa only [closure_set_of_lt_re] using eq_on.of_subset_closure this hd.continuous_on
continuous_on_const subset_closure subset.rfl },
-- Conside... | lemma | phragmen_lindelof.eq_zero_on_right_half_plane_of_superexponential_decay | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"complex.abs_pow",
"continuous_on_const",
"diff_cont_on_cl",
"diff_cont_on_cl.smul",
"exp",
"mul_le_mul",
"mul_le_mul_of_nonneg_left",
"mul_zero",
"norm_smul",
"one_lt_two",
"one_mul",
"one_pow",
"real.exp_add",
"real.exp_le_exp",
"real.exp_nat_mul",
"real.exp_pos",
... | **Phragmen-Lindelöf principle** in the right half-plane. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open right half-plane and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open right half-plane
for some `c < 2`;
* `‖f z‖` is bounded fro... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_right_half_plane_of_superexponential_decay {g : ℂ → E}
(hfd : diff_cont_on_cl ℂ f {z | 0 < z.re}) (hgd : diff_cont_on_cl ℂ g {z | 0 < z.re})
(hfexp : ∃ (c < (2 : ℝ)) B, f =O[comap complex.abs at_top ⊓ 𝓟 {z | 0 < z.re}]
(λ z, expR (B * (abs z) ^ c)))
(hgexp : ∃ (c < (2 : ℝ)) B, g =O[comap complex.abs at... | begin
suffices : eq_on (f - g) 0 {z : ℂ | 0 ≤ z.re},
by simpa only [eq_on, pi.sub_apply, pi.zero_apply, sub_eq_zero] using this,
refine eq_zero_on_right_half_plane_of_superexponential_decay (hfd.sub hgd) _ hre _,
{ set l : filter ℂ := comap complex.abs at_top ⊓ 𝓟 {z : ℂ | 0 < z.re},
suffices : ∀ {c₁ c₂ B... | lemma | phragmen_lindelof.eq_on_right_half_plane_of_superexponential_decay | analysis.complex | src/analysis/complex/phragmen_lindelof.lean | [
"analysis.complex.abs_max",
"analysis.asymptotics.superpolynomial_decay"
] | [
"complex.abs",
"diff_cont_on_cl",
"filter",
"inf_le_left",
"mul_le_mul",
"one_mul",
"real.exp_le_exp",
"real.exp_pos",
"real.norm_of_nonneg",
"real.rpow_le_rpow_of_exponent_le",
"real.rpow_nonneg_of_nonneg"
] | **Phragmen-Lindelöf principle** in the right half-plane. Let `f g : ℂ → E` be functions such
that
* `f` and `g` are differentiable in the open right half-plane and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * (abs z) ^ c)` on the open right
half-plane for some `c < 2`;
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, is_root f z | begin
contrapose! hf,
obtain ⟨c, hc⟩ := (f.differentiable.inv hf).exists_const_forall_eq_of_bounded _,
{ obtain rfl : f = C c⁻¹ := polynomial.funext (λ z, by rw [eval_C, ← hc z, inv_inv]),
exact degree_C_le },
{ obtain ⟨z₀, h₀⟩ := f.exists_forall_norm_le,
simp only [bounded_iff_forall_norm_le, set.foral... | lemma | complex.exists_root | analysis.complex | src/analysis/complex/polynomial.lean | [
"analysis.complex.liouville",
"field_theory.is_alg_closed.basic"
] | [
"inv_inv",
"inv_le_inv_of_le",
"norm_inv",
"polynomial.funext",
"set.forall_range_iff"
] | **Fundamental theorem of algebra**: every non constant complex polynomial
has a root | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_alg_closed : is_alg_closed ℂ | is_alg_closed.of_exists_root _ $ λ p _ hp, complex.exists_root $ degree_pos_of_irreducible hp | instance | complex.is_alg_closed | analysis.complex | src/analysis/complex/polynomial.lean | [
"analysis.complex.liouville",
"field_theory.is_alg_closed.basic"
] | [
"complex.exists_root",
"is_alg_closed",
"is_alg_closed.of_exists_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.real_of_complex (h : has_strict_deriv_at e e' z) :
has_strict_deriv_at (λx:ℝ, (e x).re) e'.re z | begin
have A : has_strict_fderiv_at (coe : ℝ → ℂ) of_real_clm z := of_real_clm.has_strict_fderiv_at,
have B : has_strict_fderiv_at e
((continuous_linear_map.smul_right 1 e' : ℂ →L[ℂ] ℂ).restrict_scalars ℝ)
(of_real_clm z) :=
h.has_strict_fderiv_at.restrict_scalars ℝ,
have C : has_strict_fderiv_at re r... | theorem | has_strict_deriv_at.real_of_complex | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"continuous_linear_map.smul_right",
"has_strict_deriv_at",
"has_strict_fderiv_at",
"restrict_scalars"
] | If a complex function is differentiable at a real point, then the induced real function is also
differentiable at this point, with a derivative equal to the real part of the complex derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at.real_of_complex (h : has_deriv_at e e' z) :
has_deriv_at (λx:ℝ, (e x).re) e'.re z | begin
have A : has_fderiv_at (coe : ℝ → ℂ) of_real_clm z := of_real_clm.has_fderiv_at,
have B : has_fderiv_at e ((continuous_linear_map.smul_right 1 e' : ℂ →L[ℂ] ℂ).restrict_scalars ℝ)
(of_real_clm z) :=
h.has_fderiv_at.restrict_scalars ℝ,
have C : has_fderiv_at re re_clm (e (of_real_clm z)) := re_clm.has... | theorem | has_deriv_at.real_of_complex | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"continuous_linear_map.smul_right",
"has_deriv_at",
"has_fderiv_at",
"restrict_scalars"
] | If a complex function `e` is differentiable at a real point, then the function `ℝ → ℝ` given by
the real part of `e` is also differentiable at this point, with a derivative equal to the real part
of the complex derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.real_of_complex {n : ℕ∞} (h : cont_diff_at ℂ n e z) :
cont_diff_at ℝ n (λ x : ℝ, (e x).re) z | begin
have A : cont_diff_at ℝ n (coe : ℝ → ℂ) z,
from of_real_clm.cont_diff.cont_diff_at,
have B : cont_diff_at ℝ n e z := h.restrict_scalars ℝ,
have C : cont_diff_at ℝ n re (e z), from re_clm.cont_diff.cont_diff_at,
exact C.comp z (B.comp z A)
end | theorem | cont_diff_at.real_of_complex | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.real_of_complex {n : ℕ∞} (h : cont_diff ℂ n e) :
cont_diff ℝ n (λ x : ℝ, (e x).re) | cont_diff_iff_cont_diff_at.2 $ λ x,
h.cont_diff_at.real_of_complex | theorem | cont_diff.real_of_complex | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.complex_to_real_fderiv' {f : ℂ → E} {x : ℂ} {f' : E}
(h : has_strict_deriv_at f f' x) :
has_strict_fderiv_at f (re_clm.smul_right f' + I • im_clm.smul_right f') x | by simpa only [complex.restrict_scalars_one_smul_right']
using h.has_strict_fderiv_at.restrict_scalars ℝ | lemma | has_strict_deriv_at.complex_to_real_fderiv' | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right'",
"has_strict_deriv_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.complex_to_real_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : has_deriv_at f f' x) :
has_fderiv_at f (re_clm.smul_right f' + I • im_clm.smul_right f') x | by simpa only [complex.restrict_scalars_one_smul_right']
using h.has_fderiv_at.restrict_scalars ℝ | lemma | has_deriv_at.complex_to_real_fderiv' | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right'",
"has_deriv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.complex_to_real_fderiv' {f : ℂ → E} {s : set ℂ} {x : ℂ} {f' : E}
(h : has_deriv_within_at f f' s x) :
has_fderiv_within_at f (re_clm.smul_right f' + I • im_clm.smul_right f') s x | by simpa only [complex.restrict_scalars_one_smul_right']
using h.has_fderiv_within_at.restrict_scalars ℝ | lemma | has_deriv_within_at.complex_to_real_fderiv' | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right'",
"has_deriv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.complex_to_real_fderiv {f : ℂ → ℂ} {f' x : ℂ}
(h : has_strict_deriv_at f f' x) :
has_strict_fderiv_at f (f' • (1 : ℂ →L[ℝ] ℂ)) x | by simpa only [complex.restrict_scalars_one_smul_right]
using h.has_strict_fderiv_at.restrict_scalars ℝ | lemma | has_strict_deriv_at.complex_to_real_fderiv | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right",
"has_strict_deriv_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.complex_to_real_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : has_deriv_at f f' x) :
has_fderiv_at f (f' • (1 : ℂ →L[ℝ] ℂ)) x | by simpa only [complex.restrict_scalars_one_smul_right]
using h.has_fderiv_at.restrict_scalars ℝ | lemma | has_deriv_at.complex_to_real_fderiv | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right",
"has_deriv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.complex_to_real_fderiv {f : ℂ → ℂ} {s : set ℂ} {f' x : ℂ}
(h : has_deriv_within_at f f' s x) :
has_fderiv_within_at f (f' • (1 : ℂ →L[ℝ] ℂ)) s x | by simpa only [complex.restrict_scalars_one_smul_right]
using h.has_fderiv_within_at.restrict_scalars ℝ | lemma | has_deriv_within_at.complex_to_real_fderiv | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"complex.restrict_scalars_one_smul_right",
"has_deriv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.comp_of_real (hf : has_deriv_at e e' ↑z) : has_deriv_at (λ (y:ℝ), e ↑y) e' z | by simpa only [of_real_clm_apply, of_real_one, mul_one]
using hf.comp z of_real_clm.has_deriv_at | lemma | has_deriv_at.comp_of_real | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"has_deriv_at",
"mul_one"
] | If a complex function `e` is differentiable at a real point, then its restriction to `ℝ` is
differentiable there as a function `ℝ → ℂ`, with the same derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at.of_real_comp {f : ℝ → ℝ} {u : ℝ} (hf : has_deriv_at f u z) :
has_deriv_at (λ (y:ℝ), ↑(f y) : ℝ → ℂ) u z | by simpa only [of_real_clm_apply, of_real_one, real_smul, mul_one]
using of_real_clm.has_deriv_at.scomp z hf | lemma | has_deriv_at.of_real_comp | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"has_deriv_at",
"mul_one"
] | If a function `f : ℝ → ℝ` is differentiable at a (real) point `x`, then it is also
differentiable as a function `ℝ → ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at.conformal_at (h : differentiable_at ℂ f z) (hf' : deriv f z ≠ 0) :
conformal_at f z | begin
rw [conformal_at_iff_is_conformal_map_fderiv, (h.has_fderiv_at.restrict_scalars ℝ).fderiv],
apply is_conformal_map_complex_linear,
simpa only [ne.def, ext_ring_iff]
end | lemma | differentiable_at.conformal_at | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"conformal_at",
"conformal_at_iff_is_conformal_map_fderiv",
"deriv",
"differentiable_at",
"fderiv",
"is_conformal_map_complex_linear"
] | A real differentiable function of the complex plane into some complex normed space `E` is
conformal at a point `z` if it is holomorphic at that point with a nonvanishing differential.
This is a version of the Cauchy-Riemann equations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_at_iff_differentiable_at_or_differentiable_at_comp_conj {f : ℂ → ℂ} {z : ℂ} :
conformal_at f z ↔
(differentiable_at ℂ f z ∨ differentiable_at ℂ (f ∘ conj) (conj z)) ∧ fderiv ℝ f z ≠ 0 | begin
rw conformal_at_iff_is_conformal_map_fderiv,
rw is_conformal_map_iff_is_complex_or_conj_linear,
apply and_congr_left,
intros h,
have h_diff := h.imp_symm fderiv_zero_of_not_differentiable_at,
apply or_congr,
{ rw differentiable_at_iff_restrict_scalars ℝ h_diff },
rw ← conj_conj z at h_diff,
rw d... | lemma | conformal_at_iff_differentiable_at_or_differentiable_at_comp_conj | analysis.complex | src/analysis/complex/real_deriv.lean | [
"analysis.calculus.cont_diff",
"analysis.calculus.deriv.linear",
"analysis.complex.conformal",
"analysis.calculus.conformal.normed_space"
] | [
"and_congr_left",
"conformal_at",
"conformal_at_iff_is_conformal_map_fderiv",
"differentiable_at",
"differentiable_at_iff_restrict_scalars",
"fderiv",
"fderiv.comp",
"fderiv_zero_of_not_differentiable_at",
"is_conformal_map_iff_is_complex_or_conj_linear"
] | A complex function is conformal if and only if the function is holomorphic or antiholomorphic
with a nonvanishing differential. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_at_of_differentiable_on_punctured_nhds_of_continuous_at {f : ℂ → E} {c : ℂ}
(hd : ∀ᶠ z in 𝓝[≠] c, differentiable_at ℂ f z) (hc : continuous_at f c) :
analytic_at ℂ f c | begin
rcases (nhds_within_has_basis nhds_basis_closed_ball _).mem_iff.1 hd with ⟨R, hR0, hRs⟩,
lift R to ℝ≥0 using hR0.le,
replace hc : continuous_on f (closed_ball c R),
{ refine λ z hz, continuous_at.continuous_within_at _,
rcases eq_or_ne z c with rfl | hne,
exacts [hc, (hRs ⟨hz, hne⟩).continuous_at]... | lemma | complex.analytic_at_of_differentiable_on_punctured_nhds_of_continuous_at | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"analytic_at",
"continuous_at",
"continuous_at.continuous_within_at",
"continuous_on",
"differentiable_at",
"eq_or_ne",
"lift",
"nhds_within_has_basis"
] | **Removable singularity** theorem, weak version. If `f : ℂ → E` is differentiable in a punctured
neighborhood of a point and is continuous at this point, then it is analytic at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_on_compl_singleton_and_continuous_at_iff {f : ℂ → E} {s : set ℂ} {c : ℂ}
(hs : s ∈ 𝓝 c) : differentiable_on ℂ f (s \ {c}) ∧ continuous_at f c ↔ differentiable_on ℂ f s | begin
refine ⟨_, λ hd, ⟨hd.mono (diff_subset _ _), (hd.differentiable_at hs).continuous_at⟩⟩,
rintro ⟨hd, hc⟩ x hx,
rcases eq_or_ne x c with rfl | hne,
{ refine (analytic_at_of_differentiable_on_punctured_nhds_of_continuous_at _ hc)
.differentiable_at.differentiable_within_at,
refine eventually_nhds_w... | lemma | complex.differentiable_on_compl_singleton_and_continuous_at_iff | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"continuous_at",
"differentiable_at.differentiable_within_at",
"differentiable_on",
"differentiable_within_at",
"eq_or_ne",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_dslope {f : ℂ → E} {s : set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c) :
differentiable_on ℂ (dslope f c) s ↔ differentiable_on ℂ f s | ⟨λ h, h.of_dslope, λ h, (differentiable_on_compl_singleton_and_continuous_at_iff hc).mp $
⟨iff.mpr (differentiable_on_dslope_of_nmem $ λ h, h.2 rfl) (h.mono $ diff_subset _ _),
continuous_at_dslope_same.2 $ h.differentiable_at hc⟩⟩ | lemma | complex.differentiable_on_dslope | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"differentiable_on",
"differentiable_on_dslope_of_nmem",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_update_lim_of_is_o {f : ℂ → E} {s : set ℂ} {c : ℂ}
(hc : s ∈ 𝓝 c) (hd : differentiable_on ℂ f (s \ {c}))
(ho : (λ z, f z - f c) =o[𝓝[≠] c] (λ z, (z - c)⁻¹)) :
differentiable_on ℂ (update f c (lim (𝓝[≠] c) f)) s | begin
set F : ℂ → E := λ z, (z - c) • f z with hF,
suffices : differentiable_on ℂ F (s \ {c}) ∧ continuous_at F c,
{ rw [differentiable_on_compl_singleton_and_continuous_at_iff hc, ← differentiable_on_dslope hc,
dslope_sub_smul] at this; try { apply_instance },
have hc : tendsto f (𝓝[≠] c) (𝓝 (deriv F... | lemma | complex.differentiable_on_update_lim_of_is_o | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"continuous_at",
"continuous_within_at",
"continuous_within_at_compl_self",
"deriv",
"differentiable_on",
"dslope_sub_smul",
"lim",
"smul_add",
"tendsto_const_nhds",
"update"
] | **Removable singularity** theorem: if `s` is a neighborhood of `c : ℂ`, a function `f : ℂ → E`
is complex differentiable on `s \ {c}`, and $f(z) - f(c)=o((z-c)^{-1})$, then `f` redefined to be
equal to `lim (𝓝[≠] c) f` at `c` is complex differentiable on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_on_update_lim_insert_of_is_o {f : ℂ → E} {s : set ℂ} {c : ℂ}
(hc : s ∈ 𝓝[≠] c) (hd : differentiable_on ℂ f s)
(ho : (λ z, f z - f c) =o[𝓝[≠] c] (λ z, (z - c)⁻¹)) :
differentiable_on ℂ (update f c (lim (𝓝[≠] c) f)) (insert c s) | differentiable_on_update_lim_of_is_o (insert_mem_nhds_iff.2 hc)
(hd.mono $ λ z hz, hz.1.resolve_left hz.2) ho | lemma | complex.differentiable_on_update_lim_insert_of_is_o | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"differentiable_on",
"lim",
"update"
] | **Removable singularity** theorem: if `s` is a punctured neighborhood of `c : ℂ`, a function
`f : ℂ → E` is complex differentiable on `s`, and $f(z) - f(c)=o((z-c)^{-1})$, then `f` redefined to
be equal to `lim (𝓝[≠] c) f` at `c` is complex differentiable on `{c} ∪ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_on_update_lim_of_bdd_above {f : ℂ → E} {s : set ℂ} {c : ℂ}
(hc : s ∈ 𝓝 c) (hd : differentiable_on ℂ f (s \ {c}))
(hb : bdd_above (norm ∘ f '' (s \ {c}))) :
differentiable_on ℂ (update f c (lim (𝓝[≠] c) f)) s | differentiable_on_update_lim_of_is_o hc hd $ is_bounded_under.is_o_sub_self_inv $
let ⟨C, hC⟩ := hb in ⟨C + ‖f c‖, eventually_map.2 $ mem_nhds_within_iff_exists_mem_nhds_inter.2
⟨s, hc, λ z hz, norm_sub_le_of_le (hC $ mem_image_of_mem _ hz) le_rfl⟩⟩ | lemma | complex.differentiable_on_update_lim_of_bdd_above | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"bdd_above",
"differentiable_on",
"lim",
"update"
] | **Removable singularity** theorem: if `s` is a neighborhood of `c : ℂ`, a function `f : ℂ → E`
is complex differentiable and is bounded on `s \ {c}`, then `f` redefined to be equal to
`lim (𝓝[≠] c) f` at `c` is complex differentiable on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o {f : ℂ → E} {c : ℂ}
(hd : ∀ᶠ z in 𝓝[≠] c, differentiable_at ℂ f z)
(ho : (λ z, f z - f c) =o[𝓝[≠] c] (λ z, (z - c)⁻¹)) :
tendsto f (𝓝[≠] c) (𝓝 $ lim (𝓝[≠] c) f) | begin
rw eventually_nhds_within_iff at hd,
have : differentiable_on ℂ f ({z | z ≠ c → differentiable_at ℂ f z} \ {c}),
from λ z hz, (hz.1 hz.2).differentiable_within_at,
have H := differentiable_on_update_lim_of_is_o hd this ho,
exact continuous_at_update_same.1 (H.differentiable_at hd).continuous_at
end | lemma | complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"continuous_at",
"differentiable_at",
"differentiable_on",
"differentiable_within_at",
"eventually_nhds_within_iff",
"lim"
] | **Removable singularity** theorem: if a function `f : ℂ → E` is complex differentiable on a
punctured neighborhood of `c` and $f(z) - f(c)=o((z-c)^{-1})$, then `f` has a limit at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_lim_of_differentiable_on_punctured_nhds_of_bounded_under {f : ℂ → E}
{c : ℂ} (hd : ∀ᶠ z in 𝓝[≠] c, differentiable_at ℂ f z)
(hb : is_bounded_under (≤) (𝓝[≠] c) (λ z, ‖f z - f c‖)) :
tendsto f (𝓝[≠] c) (𝓝 $ lim (𝓝[≠] c) f) | tendsto_lim_of_differentiable_on_punctured_nhds_of_is_o hd hb.is_o_sub_self_inv | lemma | complex.tendsto_lim_of_differentiable_on_punctured_nhds_of_bounded_under | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"differentiable_at",
"lim"
] | **Removable singularity** theorem: if a function `f : ℂ → E` is complex differentiable and
bounded on a punctured neighborhood of `c`, then `f` has a limit at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable
{U : set ℂ} (hU : is_open U) {c w₀ : ℂ} {R : ℝ} {f : ℂ → E}
(hc : closed_ball c R ⊆ U) (hf : differentiable_on ℂ f U) (hw₀ : w₀ ∈ ball c R) :
(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = deriv f w₀ | begin
-- We apply the removable singularity theorem and the Cauchy formula to `dslope f w₀`
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (nonempty_of_mem hw₀).ne_empty),
have hf' : differentiable_on ℂ (dslope f w₀) U,
from (differentiable_on_dslope (hU.mem_nhds ((ball_subset_closed_ball.trans hc) hw₀)))... | lemma | complex.two_pi_I_inv_smul_circle_integral_sub_sq_inv_smul_of_differentiable | analysis.complex | src/analysis/complex/removable_singularity.lean | [
"analysis.calculus.fderiv_analytic",
"analysis.asymptotics.specific_asymptotics",
"analysis.complex.cauchy_integral"
] | [
"algebra.id.smul_eq_mul",
"circle_integrable",
"circle_integral.integral_congr",
"circle_integral.integral_smul_const",
"circle_integral.integral_sub",
"circle_integral.integral_sub_zpow_of_ne",
"continuous_const",
"continuous_on",
"continuous_on.circle_integrable",
"continuous_on.inv₀",
"contin... | The Cauchy formula for the derivative of a holomorphic function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_homeomorphic_trivial_fiber_bundle_re : is_homeomorphic_trivial_fiber_bundle ℝ re | ⟨equiv_real_prod_clm.to_homeomorph, λ z, rfl⟩ | lemma | complex.is_homeomorphic_trivial_fiber_bundle_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"is_homeomorphic_trivial_fiber_bundle"
] | `complex.re` turns `ℂ` into a trivial topological fiber bundle over `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_homeomorphic_trivial_fiber_bundle_im : is_homeomorphic_trivial_fiber_bundle ℝ im | ⟨equiv_real_prod_clm.to_homeomorph.trans (homeomorph.prod_comm ℝ ℝ), λ z, rfl⟩ | lemma | complex.is_homeomorphic_trivial_fiber_bundle_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"homeomorph.prod_comm",
"is_homeomorphic_trivial_fiber_bundle"
] | `complex.im` turns `ℂ` into a trivial topological fiber bundle over `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_re : is_open_map re | is_homeomorphic_trivial_fiber_bundle_re.is_open_map_proj | lemma | complex.is_open_map_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_im : is_open_map im | is_homeomorphic_trivial_fiber_bundle_im.is_open_map_proj | lemma | complex.is_open_map_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_re : quotient_map re | is_homeomorphic_trivial_fiber_bundle_re.quotient_map_proj | lemma | complex.quotient_map_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_im : quotient_map im | is_homeomorphic_trivial_fiber_bundle_im.quotient_map_proj | lemma | complex.quotient_map_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_preimage_re (s : set ℝ) : interior (re ⁻¹' s) = re ⁻¹' (interior s) | (is_open_map_re.preimage_interior_eq_interior_preimage continuous_re _).symm | lemma | complex.interior_preimage_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_preimage_im (s : set ℝ) : interior (im ⁻¹' s) = im ⁻¹' (interior s) | (is_open_map_im.preimage_interior_eq_interior_preimage continuous_im _).symm | lemma | complex.interior_preimage_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_preimage_re (s : set ℝ) : closure (re ⁻¹' s) = re ⁻¹' (closure s) | (is_open_map_re.preimage_closure_eq_closure_preimage continuous_re _).symm | lemma | complex.closure_preimage_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_preimage_im (s : set ℝ) : closure (im ⁻¹' s) = im ⁻¹' (closure s) | (is_open_map_im.preimage_closure_eq_closure_preimage continuous_im _).symm | lemma | complex.closure_preimage_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_preimage_re (s : set ℝ) : frontier (re ⁻¹' s) = re ⁻¹' (frontier s) | (is_open_map_re.preimage_frontier_eq_frontier_preimage continuous_re _).symm | lemma | complex.frontier_preimage_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_preimage_im (s : set ℝ) : frontier (im ⁻¹' s) = im ⁻¹' (frontier s) | (is_open_map_im.preimage_frontier_eq_frontier_preimage continuous_im _).symm | lemma | complex.frontier_preimage_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_set_of_re_le (a : ℝ) : interior {z : ℂ | z.re ≤ a} = {z | z.re < a} | by simpa only [interior_Iic] using interior_preimage_re (Iic a) | lemma | complex.interior_set_of_re_le | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior",
"interior_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_set_of_im_le (a : ℝ) : interior {z : ℂ | z.im ≤ a} = {z | z.im < a} | by simpa only [interior_Iic] using interior_preimage_im (Iic a) | lemma | complex.interior_set_of_im_le | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior",
"interior_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_set_of_le_re (a : ℝ) : interior {z : ℂ | a ≤ z.re} = {z | a < z.re} | by simpa only [interior_Ici] using interior_preimage_re (Ici a) | lemma | complex.interior_set_of_le_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior",
"interior_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_set_of_le_im (a : ℝ) : interior {z : ℂ | a ≤ z.im} = {z | a < z.im} | by simpa only [interior_Ici] using interior_preimage_im (Ici a) | lemma | complex.interior_set_of_le_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior",
"interior_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_set_of_re_lt (a : ℝ) : closure {z : ℂ | z.re < a} = {z | z.re ≤ a} | by simpa only [closure_Iio] using closure_preimage_re (Iio a) | lemma | complex.closure_set_of_re_lt | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"closure_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_set_of_im_lt (a : ℝ) : closure {z : ℂ | z.im < a} = {z | z.im ≤ a} | by simpa only [closure_Iio] using closure_preimage_im (Iio a) | lemma | complex.closure_set_of_im_lt | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"closure_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_set_of_lt_re (a : ℝ) : closure {z : ℂ | a < z.re} = {z | a ≤ z.re} | by simpa only [closure_Ioi] using closure_preimage_re (Ioi a) | lemma | complex.closure_set_of_lt_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"closure_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_set_of_lt_im (a : ℝ) : closure {z : ℂ | a < z.im} = {z | a ≤ z.im} | by simpa only [closure_Ioi] using closure_preimage_im (Ioi a) | lemma | complex.closure_set_of_lt_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"closure_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_re_le (a : ℝ) : frontier {z : ℂ | z.re ≤ a} = {z | z.re = a} | by simpa only [frontier_Iic] using frontier_preimage_re (Iic a) | lemma | complex.frontier_set_of_re_le | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_im_le (a : ℝ) : frontier {z : ℂ | z.im ≤ a} = {z | z.im = a} | by simpa only [frontier_Iic] using frontier_preimage_im (Iic a) | lemma | complex.frontier_set_of_im_le | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_le_re (a : ℝ) : frontier {z : ℂ | a ≤ z.re} = {z | z.re = a} | by simpa only [frontier_Ici] using frontier_preimage_re (Ici a) | lemma | complex.frontier_set_of_le_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_le_im (a : ℝ) : frontier {z : ℂ | a ≤ z.im} = {z | z.im = a} | by simpa only [frontier_Ici] using frontier_preimage_im (Ici a) | lemma | complex.frontier_set_of_le_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_re_lt (a : ℝ) : frontier {z : ℂ | z.re < a} = {z | z.re = a} | by simpa only [frontier_Iio] using frontier_preimage_re (Iio a) | lemma | complex.frontier_set_of_re_lt | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_im_lt (a : ℝ) : frontier {z : ℂ | z.im < a} = {z | z.im = a} | by simpa only [frontier_Iio] using frontier_preimage_im (Iio a) | lemma | complex.frontier_set_of_im_lt | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_lt_re (a : ℝ) : frontier {z : ℂ | a < z.re} = {z | z.re = a} | by simpa only [frontier_Ioi] using frontier_preimage_re (Ioi a) | lemma | complex.frontier_set_of_lt_re | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_lt_im (a : ℝ) : frontier {z : ℂ | a < z.im} = {z | z.im = a} | by simpa only [frontier_Ioi] using frontier_preimage_im (Ioi a) | lemma | complex.frontier_set_of_lt_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"frontier",
"frontier_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_re_prod_im (s t : set ℝ) : closure (s ×ℂ t) = closure s ×ℂ closure t | by simpa only [← preimage_eq_preimage equiv_real_prod_clm.symm.to_homeomorph.surjective,
equiv_real_prod_clm.symm.to_homeomorph.preimage_closure]
using @closure_prod_eq _ _ _ _ s t | lemma | complex.closure_re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"closure_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_re_prod_im (s t : set ℝ) : interior (s ×ℂ t) = interior s ×ℂ interior t | by rw [re_prod_im, re_prod_im, interior_inter, interior_preimage_re, interior_preimage_im] | lemma | complex.interior_re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"interior",
"interior_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_re_prod_im (s t : set ℝ) :
frontier (s ×ℂ t) = (closure s ×ℂ frontier t) ∪ (frontier s ×ℂ closure t) | by simpa only [← preimage_eq_preimage equiv_real_prod_clm.symm.to_homeomorph.surjective,
equiv_real_prod_clm.symm.to_homeomorph.preimage_frontier]
using frontier_prod_eq s t | lemma | complex.frontier_re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure",
"frontier",
"frontier_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_le_re_and_le_im (a b : ℝ) :
frontier {z | a ≤ re z ∧ b ≤ im z} = {z | a ≤ re z ∧ im z = b ∨ re z = a ∧ b ≤ im z} | by simpa only [closure_Ici, frontier_Ici] using frontier_re_prod_im (Ici a) (Ici b) | lemma | complex.frontier_set_of_le_re_and_le_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure_Ici",
"frontier",
"frontier_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_set_of_le_re_and_im_le (a b : ℝ) :
frontier {z | a ≤ re z ∧ im z ≤ b} = {z | a ≤ re z ∧ im z = b ∨ re z = a ∧ im z ≤ b} | by simpa only [closure_Ici, closure_Iic, frontier_Ici, frontier_Iic]
using frontier_re_prod_im (Ici a) (Iic b) | lemma | complex.frontier_set_of_le_re_and_im_le | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"closure_Ici",
"closure_Iic",
"frontier",
"frontier_Ici",
"frontier_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.re_prod_im (hs : is_open s) (ht : is_open t) : is_open (s ×ℂ t) | (hs.preimage continuous_re).inter (ht.preimage continuous_im) | lemma | is_open.re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.re_prod_im (hs : is_closed s) (ht : is_closed t) : is_closed (s ×ℂ t) | (hs.preimage continuous_re).inter (ht.preimage continuous_im) | lemma | is_closed.re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.re_prod_im (hs : bounded s) (ht : bounded t) : bounded (s ×ℂ t) | antilipschitz_equiv_real_prod.bounded_preimage (hs.prod ht) | lemma | metric.bounded.re_prod_im | analysis.complex | src/analysis/complex/re_im_topology.lean | [
"analysis.complex.basic",
"topology.fiber_bundle.is_homeomorphic_trivial_bundle"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
schwarz_aux {f : ℂ → ℂ} (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : maps_to f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ | begin
have hR₁ : 0 < R₁, from nonempty_ball.1 ⟨z, hz⟩,
suffices : ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r,
{ refine ge_of_tendsto _ this,
exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhds_within_le_nhds },
rw mem_ball at hz,
filter_upwards [Ioo_mem_nhds_within_Iio ⟨hz, le_rfl⟩] with r hr,... | lemma | complex.schwarz_aux | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"Ioo_mem_nhds_within_Iio",
"closure_ball",
"diff_cont_on_cl",
"differentiable_on",
"differentiable_on.diff_cont_on_cl",
"div_eq_inv_mul",
"div_le_div_right",
"dslope",
"dslope_of_ne",
"frontier_ball",
"ge_of_tendsto",
"nhds_within_le_nhds",
"norm_inv",
"norm_smul",
"slope_def_module"
] | An auxiliary lemma for `complex.norm_dslope_le_div_of_maps_to_ball`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_dslope_le_div_of_maps_to_ball (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : maps_to f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ | begin
have hR₁ : 0 < R₁, from nonempty_ball.1 ⟨z, hz⟩,
have hR₂ : 0 < R₂, from nonempty_ball.1 ⟨f z, h_maps hz⟩,
cases eq_or_ne (dslope f c z) 0 with hc hc,
{ rw [hc, norm_zero], exact div_nonneg hR₂.le hR₁.le },
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩,
have hg' : ‖g‖₊ = 1, from nnreal.eq hg,
h... | lemma | complex.norm_dslope_le_div_of_maps_to_ball | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"abs_norm",
"differentiable_on",
"div_nonneg",
"dslope",
"eq_or_ne",
"exists_dual_vector",
"is_R_or_C.norm_of_real",
"nnreal.coe_one",
"nnreal.eq",
"one_mul",
"one_ne_zero"
] | Two cases of the **Schwarz Lemma** (derivative and distance), merged together. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_of_maps_to_ball_of_exists_norm_dslope_eq_div [complete_space E]
[strict_convex_space ℝ E] (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : set.maps_to f (ball c R₁) (ball (f c) R₂)) (h_z₀ : z₀ ∈ ball c R₁)
(h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :
set.eq_on f (λ z, f c + (z - c) • dslope f c z₀) (ball c R₁) | begin
set g := dslope f c,
rintro z hz,
by_cases z = c, { simp [h] },
have h_R₁ : 0 < R₁ := nonempty_ball.mp ⟨_, h_z₀⟩,
have g_le_div : ∀ z ∈ ball c R₁, ‖g z‖ ≤ R₂ / R₁,
from λ z hz, norm_dslope_le_div_of_maps_to_ball hd h_maps hz,
have g_max : is_max_on (norm ∘ g) (ball c R₁) z₀,
from is_max_on_iff... | lemma | complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"complete_space",
"convex_ball",
"differentiable_on",
"dslope",
"is_max_on",
"is_preconnected",
"set.eq_on",
"set.maps_to",
"strict_convex_space"
] | Equality case in the **Schwarz Lemma**: in the setup of `norm_dslope_le_div_of_maps_to_ball`, if
`‖dslope f c z₀‖ = R₂ / R₁` holds at a point in the ball then the map `f` is affine. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_of_maps_to_ball_of_exists_norm_dslope_eq_div' [complete_space E]
[strict_convex_space ℝ E] (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : set.maps_to f (ball c R₁) (ball (f c) R₂))
(h_z₀ : ∃ z₀ ∈ ball c R₁, ‖dslope f c z₀‖ = R₂ / R₁) :
∃ C : E, ‖C‖ = R₂ / R₁ ∧ set.eq_on f (λ z, f c + (z - c) • C) (ba... | let ⟨z₀, h_z₀, h_eq⟩ := h_z₀ in
⟨dslope f c z₀, h_eq, affine_of_maps_to_ball_of_exists_norm_dslope_eq_div hd h_maps h_z₀ h_eq⟩ | lemma | complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div' | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"complete_space",
"differentiable_on",
"set.eq_on",
"set.maps_to",
"strict_convex_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_deriv_le_div_of_maps_to_ball (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : maps_to f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) :
‖deriv f c‖ ≤ R₂ / R₁ | by simpa only [dslope_same] using norm_dslope_le_div_of_maps_to_ball hd h_maps (mem_ball_self h₀) | lemma | complex.norm_deriv_le_div_of_maps_to_ball | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"differentiable_on",
"dslope_same"
] | The **Schwarz Lemma**: if `f : ℂ → E` sends an open disk with center `c` and a positive radius
`R₁` to an open ball with center `f c` and radius `R₂`, then the absolute value of the derivative of
`f` at `c` is at most the ratio `R₂ / R₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_div_mul_dist_of_maps_to_ball (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : maps_to f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
dist (f z) (f c) ≤ (R₂ / R₁) * dist z c | begin
rcases eq_or_ne z c with rfl|hne, { simp only [dist_self, mul_zero] },
simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv,
← div_eq_inv_mul, ← dist_eq_norm, div_le_iff (dist_pos.2 hne)]
using norm_dslope_le_div_of_maps_to_ball hd h_maps hz
end | lemma | complex.dist_le_div_mul_dist_of_maps_to_ball | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"differentiable_on",
"dist_self",
"div_eq_inv_mul",
"div_le_iff",
"dslope_of_ne",
"eq_or_ne",
"mul_zero",
"norm_inv",
"norm_smul",
"slope_def_module"
] | The **Schwarz Lemma**: if `f : ℂ → E` sends an open disk with center `c` and radius `R₁` to an
open ball with center `f c` and radius `R₂`, then for any `z` in the former disk we have
`dist (f z) (f c) ≤ (R₂ / R₁) * dist z c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_deriv_le_div_of_maps_to_ball (hd : differentiable_on ℂ f (ball c R₁))
(h_maps : maps_to f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) :
abs (deriv f c) ≤ R₂ / R₁ | norm_deriv_le_div_of_maps_to_ball hd h_maps h₀ | lemma | complex.abs_deriv_le_div_of_maps_to_ball | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"deriv",
"differentiable_on"
] | The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk with center `c` and a positive radius
`R₁` to an open disk with center `f c` and radius `R₂`, then the absolute value of the derivative of
`f` at `c` is at most the ratio `R₂ / R₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_deriv_le_one_of_maps_to_ball (hd : differentiable_on ℂ f (ball c R))
(h_maps : maps_to f (ball c R) (ball c R)) (hc : f c = c) (h₀ : 0 < R) :
abs (deriv f c) ≤ 1 | (norm_deriv_le_div_of_maps_to_ball hd (by rwa hc) h₀).trans_eq (div_self h₀.ne') | lemma | complex.abs_deriv_le_one_of_maps_to_ball | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"deriv",
"differentiable_on",
"div_self"
] | The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk of positive radius to itself and the
center of this disk to itself, then the absolute value of the derivative of `f` at the center of
this disk is at most `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_dist_of_maps_to_ball_self (hd : differentiable_on ℂ f (ball c R))
(h_maps : maps_to f (ball c R) (ball c R)) (hc : f c = c) (hz : z ∈ ball c R) :
dist (f z) c ≤ dist z c | have hR : 0 < R, from nonempty_ball.1 ⟨z, hz⟩,
by simpa only [hc, div_self hR.ne', one_mul]
using dist_le_div_mul_dist_of_maps_to_ball hd (by rwa hc) hz | lemma | complex.dist_le_dist_of_maps_to_ball_self | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"differentiable_on",
"div_self",
"one_mul"
] | The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk to itself and the center `c` of this
disk to itself, then for any point `z` of this disk we have `dist (f z) c ≤ dist z c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_le_abs_of_maps_to_ball_self (hd : differentiable_on ℂ f (ball 0 R))
(h_maps : maps_to f (ball 0 R) (ball 0 R)) (h₀ : f 0 = 0) (hz : abs z < R) :
abs (f z) ≤ abs z | begin
replace hz : z ∈ ball (0 : ℂ) R, from mem_ball_zero_iff.2 hz,
simpa only [dist_zero_right] using dist_le_dist_of_maps_to_ball_self hd h_maps h₀ hz
end | lemma | complex.abs_le_abs_of_maps_to_ball_self | analysis.complex | src/analysis/complex/schwarz.lean | [
"analysis.complex.abs_max",
"analysis.complex.removable_singularity"
] | [
"differentiable_on"
] | The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk with center `0` to itself, the for any
point `z` of this disk we have `abs (f z) ≤ abs z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_disc : Type | ball (0 : ℂ) 1 | def | complex.unit_disc | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | Complex unit disc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_injective : injective (coe : 𝔻 → ℂ) | subtype.coe_injective | lemma | complex.unit_disc.coe_injective | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt_one (z : 𝔻) : abs (z : ℂ) < 1 | mem_ball_zero_iff.1 z.2 | lemma | complex.unit_disc.abs_lt_one | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_ne_one (z : 𝔻) : abs (z : ℂ) ≠ 1 | z.abs_lt_one.ne | lemma | complex.unit_disc.abs_ne_one | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_lt_one (z : 𝔻) : norm_sq z < 1 | @one_pow ℝ _ 2 ▸ (real.sqrt_lt' one_pos).1 z.abs_lt_one | lemma | complex.unit_disc.norm_sq_lt_one | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"one_pow",
"real.sqrt_lt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_one (z : 𝔻) : (z : ℂ) ≠ 1 | ne_of_apply_ne abs $ (map_one abs).symm ▸ z.abs_ne_one | lemma | complex.unit_disc.coe_ne_one | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"map_one",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_neg_one (z : 𝔻) : (z : ℂ) ≠ -1 | ne_of_apply_ne abs $ by { rw [abs.map_neg, map_one], exact z.abs_ne_one } | lemma | complex.unit_disc.coe_ne_neg_one | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"map_one",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_coe_ne_zero (z : 𝔻) : (1 + z : ℂ) ≠ 0 | mt neg_eq_iff_add_eq_zero.2 z.coe_ne_neg_one.symm | lemma | complex.unit_disc.one_add_coe_ne_zero | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (z w : 𝔻) : ↑(z * w) = (z * w : ℂ) | rfl | lemma | complex.unit_disc.coe_mul | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (z : ℂ) (hz : abs z < 1) : 𝔻 | ⟨z, mem_ball_zero_iff.2 hz⟩ | def | complex.unit_disc.mk | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | A constructor that assumes `abs z < 1` instead of `dist z 0 < 1` and returns an element
of `𝔻` instead of `↥metric.ball (0 : ℂ) 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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