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.instModuleSelf : Module ℂ ℂ

A shortcut instance to ensure computability; otherwise we get the noncomputable instance Complex.instNormedField.toNormedModule.toModule.

Equations

• Complex.instModuleSelf = Semiring.toModule

instance Complex.instNormedField : NormedField ℂ

Equations

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

instance Complex.instDenselyNormedField : DenselyNormedField ℂ

Equations

• Complex.instDenselyNormedField = { toNormedField := Complex.instNormedField, lt_norm_lt := Complex.instDenselyNormedField._proof_1 }

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

Equations

• Complex.instNormedAlgebraOfReal = { toAlgebra := Complex.instAlgebraOfReal, norm_smul_le := ⋯ }

@[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_intCast (n : ℤ) : ‖↑n‖₊ = ‖n‖₊

theorem Complex.continuous_normSq : Continuous ⇑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 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) : ‖equivRealProd z‖ ≤ ‖z‖

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

theorem Complex.lipschitz_equivRealProd : LipschitzWith 1 ⇑equivRealProd

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

theorem Complex.isUniformEmbedding_equivRealProd : IsUniformEmbedding ⇑equivRealProd

instance Complex.instCompleteSpace : CompleteSpace ℂ

instance Complex.instT2Space : T2Space ℂ

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_im (a✝ : ℝ × ℝ) : (equivRealProdCLM.symm a✝).im = a✝.2

@[simp]

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

@[simp]

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

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

instance Complex.instProperSpace : ProperSpace ℂ

theorem Complex.tendsto_normSq_cocompact_atTop : Filter.Tendsto (⇑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 re

theorem Complex.uniformlyContinuous_re : UniformContinuous re

@[simp]

theorem Complex.reCLM_coe : ↑reCLM = reLm

@[simp]

theorem Complex.reCLM_apply (z : ℂ) : 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 im

theorem Complex.uniformlyContinuous_im : UniformContinuous im

@[simp]

theorem Complex.imCLM_coe : ↑imCLM = imLm

@[simp]

theorem Complex.imCLM_apply (z : ℂ) : 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) = reCLM.smulRight x + I • 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.norm_conj }

@[simp]

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

@[simp]

theorem Complex.conjLIE_symm : conjLIE.symm = conjLIE

theorem Complex.isometry_conj : Isometry ⇑(starRingEnd ℂ)

@[simp]

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

@[simp]

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

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

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

instance Complex.instContinuousStar : ContinuousStar ℂ

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 : conjCLE.toLinearEquiv = conjAe.toLinearEquiv

@[simp]

theorem Complex.conjCLE_apply (z : ℂ) : 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 ofReal

theorem Complex.continuous_ofReal : Continuous ofReal

theorem Complex.isUniformEmbedding_ofReal : IsUniformEmbedding ofReal

theorem RCLike.isUniformEmbedding_ofReal {𝕜 : Type u_2} [RCLike 𝕜] : IsUniformEmbedding ofReal

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

theorem Filter.tendsto_ofReal_iff' {α : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {l : Filter α} {f : α → ℝ} {x : ℝ} : Tendsto (fun (x : α) => ↑(f x)) l (nhds ↑x) ↔ Tendsto f l (nhds x)

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

theorem Complex.ringHom_eq_ofReal_of_continuous {f : ℝ →+* ℂ} (h : Continuous ⇑f) : f = 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 : ↑ofRealCLM = ofRealAm.toLinearMap

@[simp]

theorem Complex.ofRealCLM_apply (x : ℝ) : 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 : ⇑re = Complex.re

theorem RCLike.im_eq_complex_im : ⇑im = Complex.im

theorem RCLike.ofReal_eq_complex_ofReal : ofReal = Complex.ofReal

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

theorem Complex.im_eq_zero_iff_isSelfAdjoint (x : ℂ) : x.im = 0 ↔ IsSelfAdjoint x

theorem Complex.re_eq_ofReal_of_isSelfAdjoint {x : ℂ} {y : ℝ} (hx : IsSelfAdjoint x) : x.re = y ↔ x = ↑y

theorem Complex.ofReal_eq_re_of_isSelfAdjoint {x : ℂ} {y : ℝ} (hx : IsSelfAdjoint x) : y = x.re ↔ ↑y = x

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

@[simp]

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

def RCLike.complexLinearIsometryEquiv {𝕜 : Type u_2} [RCLike 𝕜] (h : im 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_apply {𝕜 : Type u_2} [RCLike 𝕜] (h : im I = 1) (a✝ : 𝕜) : (complexLinearIsometryEquiv h) a✝ = (complexRingEquiv h).toFun a✝

@[simp]

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

theorem Complex.isometry_intCast : Isometry Int.cast

theorem Complex.closedEmbedding_intCast : Topology.IsClosedEmbedding Int.cast

theorem Complex.isClosed_range_intCast : IsClosed (Set.range Int.cast)

theorem Complex.isOpen_compl_range_intCast : IsOpen (Set.range Int.cast)ᶜ

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.

theorem Complex.norm_of_nonneg' {x : ℂ} (hx : 0 ≤ x) : ↑‖x‖ = x

theorem Complex.re_nonneg_iff_nonneg {x : ℂ} (hx : IsSelfAdjoint x) : 0 ≤ x.re ↔ 0 ≤ x

theorem Complex.re_le_re {x y : ℂ} (h : x ≤ y) : x.re ≤ y.re

@[simp]

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

@[simp]

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

@[simp]

theorem RCLike.I_to_complex : I = Complex.I

@[simp]

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

@[simp]

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

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

@[simp]

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

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

@[simp]

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

@[simp]

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

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

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

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

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

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

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

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} {L : SummationFilter α} {f : α → ℂ} {x : ℂ} : HasSum (fun (x : α) => (starRingEnd ℂ) (f x)) x L ↔ HasSum f ((starRingEnd ℂ) x) L

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

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} {L : SummationFilter α} (f : α → ℂ) : (starRingEnd ℂ) (∑'[L] (a : α), f a) = ∑'[L] (a : α), (starRingEnd ℂ) (f a)

@[simp]

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

@[simp]

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

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

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

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

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

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

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

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 ∈ slitPlane ↔ 0 < z.re ∨ z.im ≠ 0

theorem Complex.mem_slitPlane_or_neg_mem_slitPlane {z : ℂ} (hz : z ≠ 0) : z ∈ slitPlane ∨ -z ∈ slitPlane

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

theorem Complex.isOpen_slitPlane : IsOpen slitPlane

@[simp]

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

@[simp]

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

@[simp]

theorem Complex.one_mem_slitPlane : 1 ∈ slitPlane

@[simp]

theorem Complex.zero_notMem_slitPlane : 0 ∉ slitPlane

@[deprecated Complex.zero_notMem_slitPlane (since := "2025-05-23")]

theorem Complex.zero_not_mem_slitPlane : 0 ∉ slitPlane

Alias of Complex.zero_notMem_slitPlane.

@[simp]

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

@[simp]

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

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

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

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

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

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

theorem Complex.subset_slitPlane_iff_of_subset_sphere {r : ℝ} {s : Set ℂ} (hs : s ⊆ Metric.sphere 0 r) : s ⊆ slitPlane ↔ -↑r ∉ s

A subset of the circle centered at the origin in ℂ of radius r is a subset of the slitPlane if it does not contain -r.

theorem IsCompact.reProdIm {s t : Set ℝ} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s ×ℂ t)

theorem realPart.norm_le {A : Type u_1} [SeminormedAddCommGroup A] [StarAddMonoid A] [NormedSpace ℂ A] [StarModule ℂ A] [NormedStarGroup A] (x : A) : ‖realPart x‖ ≤ ‖x‖

theorem imaginaryPart.norm_le {A : Type u_1} [SeminormedAddCommGroup A] [StarAddMonoid A] [NormedSpace ℂ A] [StarModule ℂ A] [NormedStarGroup A] (x : A) : ‖imaginaryPart x‖ ≤ ‖x‖