Normed space structure on ℂ.

This file gathers basic facts of analytic nature on the complex numbers.

Main results

This file registers ℂ as a normed field, expresses basic properties of the norm, and gives tools on the real vector space structure of ℂ. Notably, it defines the following functions in the namespace Complex.

Name	Type	Description
equivRealProdCLM	ℂ ≃L[ℝ] ℝ × ℝ	The natural ContinuousLinearEquiv from ℂ to ℝ × ℝ
reCLM	ℂ →L[ℝ] ℝ	Real part function as a ContinuousLinearMap
imCLM	ℂ →L[ℝ] ℝ	Imaginary part function as a ContinuousLinearMap
ofRealCLM	ℝ →L[ℝ] ℂ	Embedding of the reals as a ContinuousLinearMap
ofRealLI	ℝ →ₗᵢ[ℝ] ℂ	Embedding of the reals as a LinearIsometry
conjCLE	ℂ ≃L[ℝ] ℂ	Complex conjugation as a ContinuousLinearEquiv
conjLIE	ℂ ≃ₗᵢ[ℝ] ℂ	Complex conjugation as a LinearIsometryEquiv

We also register the fact that ℂ is an RCLike field.

instance Complex.instNorm : Norm ℂ

Equations

• Complex.instNorm = { norm := ⇑Complex.abs }

@[simp]

theorem Complex.norm_eq_abs (z : ℂ) : ‖z‖ = Complex.abs z

theorem Complex.norm_I : ‖Complex.I‖ = 1

theorem Complex.norm_exp_ofReal_mul_I (t : ℝ) : ‖Complex.exp (↑t * Complex.I)‖ = 1

instance Complex.instNormedAddCommGroup : NormedAddCommGroup ℂ

Equations

• One or more equations did not get rendered due to their size.

instance Complex.instNormedField : NormedField ℂ

Equations

• Complex.instNormedField = NormedField.mk Complex.instNormedField.proof_1 Complex.instNormedField.proof_2

instance Complex.instDenselyNormedField : DenselyNormedField ℂ

Equations

• Complex.instDenselyNormedField = DenselyNormedField.mk Complex.instDenselyNormedField.proof_1

instance Complex.instNormedAlgebraOfReal {R : Type u_1} [NormedField R] [NormedAlgebra R ℝ] : NormedAlgebra R ℂ

Equations

• Complex.instNormedAlgebraOfReal = NormedAlgebra.mk ⋯

@[instance 900]

instance NormedSpace.complexToReal {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℂ E] : NormedSpace ℝ E

The module structure from Module.complexToReal is a normed space.

Equations

• NormedSpace.complexToReal = NormedSpace.restrictScalars ℝ ℂ E

@[instance 900]

instance NormedAlgebra.complexToReal {A : Type u_2} [SeminormedRing A] [NormedAlgebra ℂ A] : NormedAlgebra ℝ A

The algebra structure from Algebra.complexToReal is a normed algebra.

Equations

• NormedAlgebra.complexToReal = NormedAlgebra.restrictScalars ℝ ℂ A

@[simp]

theorem Complex.nnnorm_I : ‖Complex.I‖₊ = 1

theorem Complex.dist_eq (z : ℂ) (w : ℂ) : dist z w = Complex.abs (z - w)

theorem Complex.dist_eq_re_im (z : ℂ) (w : ℂ) : dist z w = √((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2)

@[simp]

theorem Complex.dist_mk (x₁ : ℝ) (y₁ : ℝ) (x₂ : ℝ) (y₂ : ℝ) : dist { re := x₁, im := y₁ } { re := x₂, im := y₂ } = √((x₁ - x₂) ^ 2 + (y₁ - y₂) ^ 2)

theorem Complex.dist_of_re_eq {z : ℂ} {w : ℂ} (h : z.re = w.re) : dist z w = dist z.im w.im

theorem Complex.nndist_of_re_eq {z : ℂ} {w : ℂ} (h : z.re = w.re) : nndist z w = nndist z.im w.im

theorem Complex.edist_of_re_eq {z : ℂ} {w : ℂ} (h : z.re = w.re) : edist z w = edist z.im w.im

theorem Complex.dist_of_im_eq {z : ℂ} {w : ℂ} (h : z.im = w.im) : dist z w = dist z.re w.re

theorem Complex.nndist_of_im_eq {z : ℂ} {w : ℂ} (h : z.im = w.im) : nndist z w = nndist z.re w.re

theorem Complex.edist_of_im_eq {z : ℂ} {w : ℂ} (h : z.im = w.im) : edist z w = edist z.re w.re

theorem Complex.dist_conj_self (z : ℂ) : dist ((starRingEnd ℂ) z) z = 2 * |z.im|

theorem Complex.nndist_conj_self (z : ℂ) : nndist ((starRingEnd ℂ) z) z = 2 * Real.nnabs z.im

theorem Complex.dist_self_conj (z : ℂ) : dist z ((starRingEnd ℂ) z) = 2 * |z.im|

theorem Complex.nndist_self_conj (z : ℂ) : nndist z ((starRingEnd ℂ) z) = 2 * Real.nnabs z.im

@[simp]

theorem Complex.comap_abs_nhds_zero : Filter.comap (⇑Complex.abs) (nhds 0) = nhds 0

@[simp]

theorem Complex.norm_real (r : ℝ) : ‖↑r‖ = ‖r‖

@[simp]

theorem Complex.nnnorm_real (r : ℝ) : ‖↑r‖₊ = ‖r‖₊

@[simp]

theorem Complex.norm_natCast (n : ℕ) : ‖↑n‖ = ↑n

@[simp]

theorem Complex.norm_intCast (n : ℤ) : ‖↑n‖ = |↑n|

@[simp]

theorem Complex.norm_ratCast (q : ℚ) : ‖↑q‖ = |↑q|

@[simp]

theorem Complex.nnnorm_natCast (n : ℕ) : ‖↑n‖₊ = ↑n

@[simp]

theorem Complex.nnnorm_intCast (n : ℤ) : ‖↑n‖₊ = ‖n‖₊

@[simp]

theorem Complex.nnnorm_ratCast (q : ℚ) : ‖↑q‖₊ = ‖↑q‖₊

@[simp]

theorem Complex.norm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖OfNat.ofNat n‖ = OfNat.ofNat n

@[simp]

theorem Complex.nnnorm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖OfNat.ofNat n‖₊ = OfNat.ofNat n

@[deprecated Complex.norm_natCast]

theorem Complex.norm_nat (n : ℕ) : ‖↑n‖ = ↑n

Alias of Complex.norm_natCast.

@[deprecated Complex.norm_intCast]

theorem Complex.norm_int (n : ℤ) : ‖↑n‖ = |↑n|

Alias of Complex.norm_intCast.

@[deprecated Complex.norm_ratCast]

theorem Complex.norm_rat (q : ℚ) : ‖↑q‖ = |↑q|

Alias of Complex.norm_ratCast.

@[deprecated Complex.nnnorm_natCast]

theorem Complex.nnnorm_nat (n : ℕ) : ‖↑n‖₊ = ↑n

Alias of Complex.nnnorm_natCast.

@[deprecated Complex.nnnorm_intCast]

theorem Complex.nnnorm_int (n : ℤ) : ‖↑n‖₊ = ‖n‖₊

Alias of Complex.nnnorm_intCast.

@[simp]

theorem Complex.norm_nnratCast (q : ℚ≥0) : ‖↑q‖ = ↑q

@[simp]

theorem Complex.nnnorm_nnratCast (q : ℚ≥0) : ‖↑q‖₊ = ↑q

theorem Complex.norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖↑n‖ = ↑n

theorem Complex.normSq_eq_norm_sq (z : ℂ) : Complex.normSq z = ‖z‖ ^ 2

theorem Complex.continuous_abs : Continuous ⇑Complex.abs

theorem Complex.continuous_normSq : Continuous ⇑Complex.normSq

theorem Complex.nnnorm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖₊ = 1

theorem Complex.norm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖ = 1

theorem Complex.le_of_eq_sum_of_eq_sum_norm {ι : Type u_2} {a : ℝ} {b : ℝ} (f : ι → ℂ) (s : Finset ι) (ha₀ : 0 ≤ a) (ha : ↑a = ∑ i ∈ s, f i) (hb : ↑b = ∑ i ∈ s, ↑‖f i‖) : a ≤ b

theorem Complex.equivRealProd_apply_le (z : ℂ) : ‖Complex.equivRealProd z‖ ≤ Complex.abs z

theorem Complex.equivRealProd_apply_le' (z : ℂ) : ‖Complex.equivRealProd z‖ ≤ 1 * Complex.abs z

theorem Complex.lipschitz_equivRealProd : LipschitzWith 1 ⇑Complex.equivRealProd

theorem Complex.antilipschitz_equivRealProd : AntilipschitzWith (NNReal.sqrt 2) ⇑Complex.equivRealProd

theorem Complex.isUniformEmbedding_equivRealProd : IsUniformEmbedding ⇑Complex.equivRealProd

@[deprecated Complex.isUniformEmbedding_equivRealProd]

theorem Complex.uniformEmbedding_equivRealProd : IsUniformEmbedding ⇑Complex.equivRealProd

Alias of Complex.isUniformEmbedding_equivRealProd.

instance Complex.instCompleteSpace : CompleteSpace ℂ

Equations

• Complex.instCompleteSpace = ⋯

instance Complex.instT2Space : T2Space ℂ

Equations

• Complex.instT2Space = ⋯

def Complex.equivRealProdCLM : ℂ ≃L[ℝ] ℝ × ℝ

The natural ContinuousLinearEquiv from ℂ to ℝ × ℝ.

Equations

• Complex.equivRealProdCLM = Complex.equivRealProdLm.toContinuousLinearEquivOfBounds 1 (√2) Complex.equivRealProd_apply_le' Complex.equivRealProdCLM.proof_2

@[simp]

theorem Complex.equivRealProdCLM_symm_apply_re : ∀ (a : ℝ × ℝ), (Complex.equivRealProdCLM.symm a).re = a.1

@[simp]

theorem Complex.equivRealProdCLM_apply : ∀ (a : ℂ), Complex.equivRealProdCLM a = (a.re, a.im)

@[simp]

theorem Complex.equivRealProdCLM_symm_apply_im : ∀ (a : ℝ × ℝ), (Complex.equivRealProdCLM.symm a).im = a.2

theorem Complex.equivRealProdCLM_symm_apply (p : ℝ × ℝ) : Complex.equivRealProdCLM.symm p = ↑p.1 + ↑p.2 * Complex.I

instance Complex.instProperSpace : ProperSpace ℂ

Equations

• Complex.instProperSpace = ⋯

theorem Complex.tendsto_abs_cocompact_atTop : Filter.Tendsto (⇑Complex.abs) (Filter.cocompact ℂ) Filter.atTop

The abs function on ℂ is proper.

theorem Complex.tendsto_normSq_cocompact_atTop : Filter.Tendsto (⇑Complex.normSq) (Filter.cocompact ℂ) Filter.atTop

The normSq function on ℂ is proper.

def Complex.reCLM : ℂ →L[ℝ] ℝ

Continuous linear map version of the real part function, from ℂ to ℝ.

Equations

• Complex.reCLM = Complex.reLm.mkContinuous 1 Complex.reCLM.proof_1

theorem Complex.continuous_re : Continuous Complex.re

theorem Complex.uniformlyContinuous_re : UniformContinuous Complex.re

@[deprecated Complex.uniformlyContinuous_re]

theorem Complex.uniformlyContinous_re : UniformContinuous Complex.re

Alias of Complex.uniformlyContinuous_re.

@[simp]

theorem Complex.reCLM_coe : ↑Complex.reCLM = Complex.reLm

@[simp]

theorem Complex.reCLM_apply (z : ℂ) : Complex.reCLM z = z.re

def Complex.imCLM : ℂ →L[ℝ] ℝ

Continuous linear map version of the imaginary part function, from ℂ to ℝ.

Equations

• Complex.imCLM = Complex.imLm.mkContinuous 1 Complex.imCLM.proof_1

theorem Complex.continuous_im : Continuous Complex.im

theorem Complex.uniformlyContinuous_im : UniformContinuous Complex.im

@[deprecated Complex.uniformlyContinuous_im]

theorem Complex.uniformlyContinous_im : UniformContinuous Complex.im

Alias of Complex.uniformlyContinuous_im.

@[simp]

theorem Complex.imCLM_coe : ↑Complex.imCLM = Complex.imLm

@[simp]

theorem Complex.imCLM_apply (z : ℂ) : Complex.imCLM z = z.im

theorem Complex.restrictScalars_one_smulRight' {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℂ E] (x : E) : ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 x) = Complex.reCLM.smulRight x + Complex.I • Complex.imCLM.smulRight x

theorem Complex.restrictScalars_one_smulRight (x : ℂ) : ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 x) = x • 1

def Complex.conjLIE : ℂ ≃ₗᵢ[ℝ] ℂ

The complex-conjugation function from ℂ to itself is an isometric linear equivalence.

Equations

• Complex.conjLIE = { toLinearEquiv := Complex.conjAe.toLinearEquiv, norm_map' := Complex.abs_conj }

@[simp]

theorem Complex.conjLIE_apply (z : ℂ) : Complex.conjLIE z = (starRingEnd ℂ) z

@[simp]

theorem Complex.conjLIE_symm : Complex.conjLIE.symm = Complex.conjLIE

theorem Complex.isometry_conj : Isometry ⇑(starRingEnd ℂ)

@[simp]

theorem Complex.dist_conj_conj (z : ℂ) (w : ℂ) : dist ((starRingEnd ℂ) z) ((starRingEnd ℂ) w) = dist z w

@[simp]

theorem Complex.nndist_conj_conj (z : ℂ) (w : ℂ) : nndist ((starRingEnd ℂ) z) ((starRingEnd ℂ) w) = nndist z w

theorem Complex.dist_conj_comm (z : ℂ) (w : ℂ) : dist ((starRingEnd ℂ) z) w = dist z ((starRingEnd ℂ) w)

theorem Complex.nndist_conj_comm (z : ℂ) (w : ℂ) : nndist ((starRingEnd ℂ) z) w = nndist z ((starRingEnd ℂ) w)

instance Complex.instContinuousStar : ContinuousStar ℂ

Equations

• Complex.instContinuousStar = ⋯

theorem Complex.continuous_conj : Continuous ⇑(starRingEnd ℂ)

theorem Complex.ringHom_eq_id_or_conj_of_continuous {f : ℂ →+* ℂ} (hf : Continuous ⇑f) : f = RingHom.id ℂ ∨ f = starRingEnd ℂ

The only continuous ring homomorphisms from ℂ to ℂ are the identity and the complex conjugation.

def Complex.conjCLE : ℂ ≃L[ℝ] ℂ

Continuous linear equiv version of the conj function, from ℂ to ℂ.

Equations

• Complex.conjCLE = { toLinearEquiv := Complex.conjLIE.toLinearEquiv, continuous_toFun := Complex.conjCLE.proof_1, continuous_invFun := Complex.conjCLE.proof_2 }

@[simp]

theorem Complex.conjCLE_coe : Complex.conjCLE.toLinearEquiv = Complex.conjAe.toLinearEquiv

@[simp]

theorem Complex.conjCLE_apply (z : ℂ) : Complex.conjCLE z = (starRingEnd ℂ) z

def Complex.ofRealLI : ℝ →ₗᵢ[ℝ] ℂ

Linear isometry version of the canonical embedding of ℝ in ℂ.

Equations

• Complex.ofRealLI = { toLinearMap := Complex.ofRealAm.toLinearMap, norm_map' := Complex.norm_real }

theorem Complex.isometry_ofReal : Isometry Complex.ofReal

theorem Complex.continuous_ofReal : Continuous Complex.ofReal

theorem Complex.isUniformEmbedding_ofReal : IsUniformEmbedding Complex.ofReal

theorem Filter.tendsto_ofReal_iff {α : Type u_2} {l : Filter α} {f : α → ℝ} {x : ℝ} : Filter.Tendsto (fun (x : α) => ↑(f x)) l (nhds ↑x) ↔ Filter.Tendsto f l (nhds x)

theorem Filter.Tendsto.ofReal {α : Type u_2} {l : Filter α} {f : α → ℝ} {x : ℝ} (hf : Filter.Tendsto f l (nhds x)) : Filter.Tendsto (fun (x : α) => ↑(f x)) l (nhds ↑x)

theorem Complex.ringHom_eq_ofReal_of_continuous {f : ℝ →+* ℂ} (h : Continuous ⇑f) : f = Complex.ofRealHom

The only continuous ring homomorphism from ℝ to ℂ is the identity.

def Complex.ofRealCLM : ℝ →L[ℝ] ℂ

Continuous linear map version of the canonical embedding of ℝ in ℂ.

Equations

• Complex.ofRealCLM = Complex.ofRealLI.toContinuousLinearMap

@[simp]

theorem Complex.ofRealCLM_coe : ↑Complex.ofRealCLM = Complex.ofRealAm.toLinearMap

@[simp]

theorem Complex.ofRealCLM_apply (x : ℝ) : Complex.ofRealCLM x = ↑x

noncomputable instance Complex.instRCLike : RCLike ℂ

Equations

• One or more equations did not get rendered due to their size.

theorem RCLike.re_eq_complex_re : ⇑RCLike.re = Complex.re

theorem RCLike.im_eq_complex_im : ⇑RCLike.im = Complex.im

theorem Complex.mul_conj' (z : ℂ) : z * (starRingEnd ℂ) z = ↑‖z‖ ^ 2

theorem Complex.conj_mul' (z : ℂ) : (starRingEnd ℂ) z * z = ↑‖z‖ ^ 2

theorem Complex.inv_eq_conj {z : ℂ} (hz : ‖z‖ = 1) : z⁻¹ = (starRingEnd ℂ) z

theorem Complex.exists_norm_eq_mul_self (z : ℂ) : ∃ (c : ℂ), ‖c‖ = 1 ∧ ↑‖z‖ = c * z

theorem Complex.exists_norm_mul_eq_self (z : ℂ) : ∃ (c : ℂ), ‖c‖ = 1 ∧ c * ↑‖z‖ = z

def RCLike.complexRingEquiv {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : 𝕜 ≃+* ℂ

The natural isomorphism between 𝕜 satisfying RCLike 𝕜 and ℂ when RCLike.im RCLike.I = 1.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem RCLike.complexRingEquiv_symm_apply {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) (x : ℂ) : (RCLike.complexRingEquiv h).symm x = ↑x.re + ↑x.im * RCLike.I

@[simp]

theorem RCLike.complexRingEquiv_apply {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) (x : 𝕜) : (RCLike.complexRingEquiv h) x = ↑(RCLike.re x) + ↑(RCLike.im x) * Complex.I

def RCLike.complexLinearIsometryEquiv {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : 𝕜 ≃ₗᵢ[ℝ] ℂ

The natural ℝ-linear isometry equivalence between 𝕜 satisfying RCLike 𝕜 and ℂ when RCLike.im RCLike.I = 1.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem RCLike.complexLinearIsometryEquiv_toFun {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : ∀ (a : 𝕜), (RCLike.complexLinearIsometryEquiv h) a = (RCLike.complexRingEquiv h).toFun a

@[simp]

theorem RCLike.complexLinearIsometryEquiv_apply {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : ∀ (a : 𝕜), (RCLike.complexLinearIsometryEquiv h) a = (RCLike.complexRingEquiv h).toFun a

@[simp]

theorem RCLike.complexLinearIsometryEquiv_symm_apply {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : ∀ (a : ℂ), (RCLike.complexLinearIsometryEquiv h).symm a = (RCLike.complexRingEquiv h).invFun a

@[simp]

theorem RCLike.complexLinearIsometryEquiv_invFun {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) : ∀ (a : ℂ), (RCLike.complexLinearIsometryEquiv h).invFun a = (RCLike.complexRingEquiv h).invFun a

theorem Complex.eq_coe_norm_of_nonneg {z : ℂ} (hz : 0 ≤ z) : z = ↑‖z‖

theorem Complex.orderClosedTopology : OrderClosedTopology ℂ

We show that the partial order and the topology on ℂ are compatible. We turn this into an instance scoped to ComplexOrder.

@[simp]

theorem RCLike.re_to_complex {x : ℂ} : RCLike.re x = x.re

@[simp]

theorem RCLike.im_to_complex {x : ℂ} : RCLike.im x = x.im

@[simp]

theorem RCLike.I_to_complex : RCLike.I = Complex.I

@[simp]

theorem RCLike.normSq_to_complex {x : ℂ} : RCLike.normSq x = Complex.normSq x

@[simp]

theorem RCLike.hasSum_conj {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} : HasSum (fun (x : α) => (starRingEnd 𝕜) (f x)) x ↔ HasSum f ((starRingEnd 𝕜) x)

theorem RCLike.hasSum_conj' {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} : HasSum (fun (x : α) => (starRingEnd 𝕜) (f x)) ((starRingEnd 𝕜) x) ↔ HasSum f x

@[simp]

theorem RCLike.summable_conj {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} : (Summable fun (x : α) => (starRingEnd 𝕜) (f x)) ↔ Summable f

theorem RCLike.conj_tsum {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] (f : α → 𝕜) : (starRingEnd 𝕜) (∑' (a : α), f a) = ∑' (a : α), (starRingEnd 𝕜) (f a)

@[simp]

theorem RCLike.hasSum_ofReal {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → ℝ} {x : ℝ} : HasSum (fun (x : α) => ↑(f x)) ↑x ↔ HasSum f x

@[simp]

theorem RCLike.summable_ofReal {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → ℝ} : (Summable fun (x : α) => ↑(f x)) ↔ Summable f

theorem RCLike.ofReal_tsum {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] (f : α → ℝ) : ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)

theorem RCLike.hasSum_re {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} (h : HasSum f x) : HasSum (fun (x : α) => RCLike.re (f x)) (RCLike.re x)

theorem RCLike.hasSum_im {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} (h : HasSum f x) : HasSum (fun (x : α) => RCLike.im (f x)) (RCLike.im x)

theorem RCLike.re_tsum {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} (h : Summable f) : RCLike.re (∑' (a : α), f a) = ∑' (a : α), RCLike.re (f a)

theorem RCLike.im_tsum {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} (h : Summable f) : RCLike.im (∑' (a : α), f a) = ∑' (a : α), RCLike.im (f a)

theorem RCLike.hasSum_iff {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] (f : α → 𝕜) (c : 𝕜) : HasSum f c ↔ HasSum (fun (x : α) => RCLike.re (f x)) (RCLike.re c) ∧ HasSum (fun (x : α) => RCLike.im (f x)) (RCLike.im c)

We have to repeat the lemmas about RCLike.re and RCLike.im as they are not syntactic matches for Complex.re and Complex.im.

We do not have this problem with ofReal and conj, although we repeat them anyway for discoverability and to avoid the need to unify 𝕜.

theorem Complex.hasSum_conj {α : Type u_1} {f : α → ℂ} {x : ℂ} : HasSum (fun (x : α) => (starRingEnd ℂ) (f x)) x ↔ HasSum f ((starRingEnd ℂ) x)

theorem Complex.hasSum_conj' {α : Type u_1} {f : α → ℂ} {x : ℂ} : HasSum (fun (x : α) => (starRingEnd ℂ) (f x)) ((starRingEnd ℂ) x) ↔ HasSum f x

theorem Complex.summable_conj {α : Type u_1} {f : α → ℂ} : (Summable fun (x : α) => (starRingEnd ℂ) (f x)) ↔ Summable f

theorem Complex.conj_tsum {α : Type u_1} (f : α → ℂ) : (starRingEnd ℂ) (∑' (a : α), f a) = ∑' (a : α), (starRingEnd ℂ) (f a)

@[simp]

theorem Complex.hasSum_ofReal {α : Type u_1} {f : α → ℝ} {x : ℝ} : HasSum (fun (x : α) => ↑(f x)) ↑x ↔ HasSum f x

@[simp]

theorem Complex.summable_ofReal {α : Type u_1} {f : α → ℝ} : (Summable fun (x : α) => ↑(f x)) ↔ Summable f

theorem Complex.ofReal_tsum {α : Type u_1} (f : α → ℝ) : ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)

theorem Complex.hasSum_re {α : Type u_1} {f : α → ℂ} {x : ℂ} (h : HasSum f x) : HasSum (fun (x : α) => (f x).re) x.re

theorem Complex.hasSum_im {α : Type u_1} {f : α → ℂ} {x : ℂ} (h : HasSum f x) : HasSum (fun (x : α) => (f x).im) x.im

theorem Complex.re_tsum {α : Type u_1} {f : α → ℂ} (h : Summable f) : (∑' (a : α), f a).re = ∑' (a : α), (f a).re

theorem Complex.im_tsum {α : Type u_1} {f : α → ℂ} (h : Summable f) : (∑' (a : α), f a).im = ∑' (a : α), (f a).im

theorem Complex.hasSum_iff {α : Type u_1} (f : α → ℂ) (c : ℂ) : HasSum f c ↔ HasSum (fun (x : α) => (f x).re) c.re ∧ HasSum (fun (x : α) => (f x).im) c.im

Define the "slit plane" ℂ ∖ ℝ≤0 and provide some API

def Complex.slitPlane : Set ℂ

The slit plane is the complex plane with the closed negative real axis removed.

Equations

• Complex.slitPlane = {z : ℂ | 0 < z.re ∨ z.im ≠ 0}

theorem Complex.mem_slitPlane_iff {z : ℂ} : z ∈ Complex.slitPlane ↔ 0 < z.re ∨ z.im ≠ 0

theorem Complex.slitPlane_eq_union : Complex.slitPlane = {z : ℂ | 0 < z.re} ∪ {z : ℂ | z.im ≠ 0}

theorem Complex.isOpen_slitPlane : IsOpen Complex.slitPlane

@[simp]

theorem Complex.ofReal_mem_slitPlane {x : ℝ} : ↑x ∈ Complex.slitPlane ↔ 0 < x

@[simp]

theorem Complex.neg_ofReal_mem_slitPlane {x : ℝ} : -↑x ∈ Complex.slitPlane ↔ x < 0

@[simp]

theorem Complex.one_mem_slitPlane : 1 ∈ Complex.slitPlane

@[simp]

theorem Complex.zero_not_mem_slitPlane : 0 ∉ Complex.slitPlane

@[simp]

theorem Complex.natCast_mem_slitPlane {n : ℕ} : ↑n ∈ Complex.slitPlane ↔ n ≠ 0

@[deprecated Complex.natCast_mem_slitPlane]

theorem Complex.nat_cast_mem_slitPlane {n : ℕ} : ↑n ∈ Complex.slitPlane ↔ n ≠ 0

Alias of Complex.natCast_mem_slitPlane.

@[simp]

theorem Complex.ofNat_mem_slitPlane (n : ℕ) [n.AtLeastTwo] : OfNat.ofNat n ∈ Complex.slitPlane

theorem Complex.mem_slitPlane_iff_not_le_zero {z : ℂ} : z ∈ Complex.slitPlane ↔ ¬z ≤ 0

theorem Complex.compl_Iic_zero : (Set.Iic 0)ᶜ = Complex.slitPlane

theorem Complex.slitPlane_ne_zero {z : ℂ} (hz : z ∈ Complex.slitPlane) : z ≠ 0

theorem Complex.ball_one_subset_slitPlane : Metric.ball 1 1 ⊆ Complex.slitPlane

The slit plane includes the open unit ball of radius 1 around 1.

theorem Complex.mem_slitPlane_of_norm_lt_one {z : ℂ} (hz : ‖z‖ < 1) : 1 + z ∈ Complex.slitPlane

The slit plane includes the open unit ball of radius 1 around 1.