The ψ Functions #

In this file, we define the functions ψI, ψT and ψS that are defined using the Jacobi theta functions and are used in the definition of the -1-eigenfunction b.

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def ModularGroup.I :

Matrix.SpecialLinearGroup (Fin 2) ℤ

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ModularGroup.I = ⟨!![1, 0; 0, 1], ModularGroup.I._proof_1 ⟩

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def h :

UpperHalfPlane → ℂ

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h = 128 • (⇑H₃_MF + ⇑H₄_MF) / ⇑H₂_MF ^ 2

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def ψI :

UpperHalfPlane → ℂ

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ψI = h - SlashAction.map (-2) (ModularGroup.S * ModularGroup.T) h

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def ψT :

UpperHalfPlane → ℂ

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ψT = SlashAction.map (-2) ModularGroup.T ψI

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def ψS :

UpperHalfPlane → ℂ

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ψS = SlashAction.map (-2) ModularGroup.S ψI

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def ψI' (z : ℂ) :

ℂ

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ψI' z = if hz : 0 < z.im then ψI { coe := z, coe_im_pos := hz } else 0

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def ψS' (z : ℂ) :

ℂ

Equations

ψS' z = if hz : 0 < z.im then ψS { coe := z, coe_im_pos := hz } else 0

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def ψT' (z : ℂ) :

ℂ

Equations

ψT' z = if hz : 0 < z.im then ψT { coe := z, coe_im_pos := hz } else 0

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theorem ψI_eq :

ψI = 128 • ((⇑H₃_MF + ⇑H₄_MF) / ⇑H₂_MF ^ 2 + (⇑H₄_MF - ⇑H₂_MF) / ⇑H₃_MF ^ 2)

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theorem ψT_eq :

ψT = 128 * ((⇑H₃_MF + ⇑H₄_MF) / ⇑H₂_MF ^ 2 + (⇑H₂_MF + ⇑H₃_MF) / ⇑H₄_MF ^ 2)

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theorem ψS_eq' :

ψS = 128 * ((⇑H₄_MF - ⇑H₂_MF) / ⇑H₃_MF ^ 2 - (⇑H₂_MF + ⇑H₃_MF) / ⇑H₄_MF ^ 2)

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theorem ψS_eq :

ψS = 128 * (-((⇑H₂_MF + ⇑H₃_MF) / ⇑H₄_MF ^ 2) - (⇑H₂_MF - ⇑H₄_MF) / ⇑H₃_MF ^ 2)

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theorem ψT_slash_T :

SlashAction.map (-2) ModularGroup.T ψT = ψI

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theorem ψS_slash_S :

SlashAction.map (-2) ModularGroup.S ψS = ψI

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theorem ψS_slash_ST :

SlashAction.map (-2) (ModularGroup.S * ModularGroup.T) ψS = ψT

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theorem ψS_slash_T :

SlashAction.map (-2) ModularGroup.T ψS = -ψS

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theorem ψT_slash_S :

SlashAction.map (-2) ModularGroup.S ψT = -ψT

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theorem ψI_slash_TS :

SlashAction.map (-2) (ModularGroup.T * ModularGroup.S) ψI = -ψT

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theorem ψS_slash_STS :

SlashAction.map (-2) (ModularGroup.S * ModularGroup.T * ModularGroup.S) ψS = -ψT

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theorem ψS_slash_TSTS :

SlashAction.map (-2) (ModularGroup.T * ModularGroup.S * ModularGroup.T * ModularGroup.S) ψS = ψT

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theorem ψI'_eq_ψI_of_mem {z : ℂ} (hz : 0 < z.im) :

ψI' z = ψI { coe := z, coe_im_pos := hz }

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theorem ψS'_eq_ψS_of_mem {z : ℂ} (hz : 0 < z.im) :

ψS' z = ψS { coe := z, coe_im_pos := hz }

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theorem ψT'_eq_ψT_of_mem {z : ℂ} (hz : 0 < z.im) :

ψT' z = ψT { coe := z, coe_im_pos := hz }

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theorem ψT'_comp_z₁'_eq_ψT_comp_z₁'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψT' (MagicFunction.Parametrisations.z₁' t) = ψT { coe := MagicFunction.Parametrisations.z₁' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₁'_eq_ψS_comp_z₁'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψS' (MagicFunction.Parametrisations.z₁' t) = ψS { coe := MagicFunction.Parametrisations.z₁' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₁'_eq_ψI_comp_z₁'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψI' (MagicFunction.Parametrisations.z₁' t) = ψI { coe := MagicFunction.Parametrisations.z₁' t, coe_im_pos := ⋯ }

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theorem ψT'_comp_z₂'_eq_ψT_comp_z₂'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψT' (MagicFunction.Parametrisations.z₂' t) = ψT { coe := MagicFunction.Parametrisations.z₂' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₂'_eq_ψS_comp_z₂'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψS' (MagicFunction.Parametrisations.z₂' t) = ψS { coe := MagicFunction.Parametrisations.z₂' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₂'_eq_ψI_comp_z₂'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψI' (MagicFunction.Parametrisations.z₂' t) = ψI { coe := MagicFunction.Parametrisations.z₂' t, coe_im_pos := ⋯ }

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theorem ψT'_comp_z₃'_eq_ψT_comp_z₃'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψT' (MagicFunction.Parametrisations.z₃' t) = ψT { coe := MagicFunction.Parametrisations.z₃' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₃'_eq_ψS_comp_z₃'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψS' (MagicFunction.Parametrisations.z₃' t) = ψS { coe := MagicFunction.Parametrisations.z₃' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₃'_eq_ψI_comp_z₃'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψI' (MagicFunction.Parametrisations.z₃' t) = ψI { coe := MagicFunction.Parametrisations.z₃' t, coe_im_pos := ⋯ }

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theorem ψT'_comp_z₄'_eq_ψT_comp_z₄'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψT' (MagicFunction.Parametrisations.z₄' t) = ψT { coe := MagicFunction.Parametrisations.z₄' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₄'_eq_ψS_comp_z₄'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψS' (MagicFunction.Parametrisations.z₄' t) = ψS { coe := MagicFunction.Parametrisations.z₄' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₄'_eq_ψI_comp_z₄'_of_mem {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψI' (MagicFunction.Parametrisations.z₄' t) = ψI { coe := MagicFunction.Parametrisations.z₄' t, coe_im_pos := ⋯ }

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theorem ψT'_comp_z₅'_eq_ψT_comp_z₅'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψT' (MagicFunction.Parametrisations.z₅' t) = ψT { coe := MagicFunction.Parametrisations.z₅' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₅'_eq_ψS_comp_z₅'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψS' (MagicFunction.Parametrisations.z₅' t) = ψS { coe := MagicFunction.Parametrisations.z₅' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₅'_eq_ψI_comp_z₅'_of_mem {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψI' (MagicFunction.Parametrisations.z₅' t) = ψI { coe := MagicFunction.Parametrisations.z₅' t, coe_im_pos := ⋯ }

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theorem ψT'_comp_z₆'_eq_ψT_comp_z₆'_of_mem {t : ℝ} (ht : t ∈ Set.Ici 1) :

ψT' (MagicFunction.Parametrisations.z₆' t) = ψT { coe := MagicFunction.Parametrisations.z₆' t, coe_im_pos := ⋯ }

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theorem ψS'_comp_z₆'_eq_ψS_comp_z₆'_of_mem {t : ℝ} (ht : t ∈ Set.Ici 1) :

ψS' (MagicFunction.Parametrisations.z₆' t) = ψS { coe := MagicFunction.Parametrisations.z₆' t, coe_im_pos := ⋯ }

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theorem ψI'_comp_z₆'_eq_ψI_comp_z₆'_of_mem {t : ℝ} (ht : t ∈ Set.Ici 1) :

ψI' (MagicFunction.Parametrisations.z₆' t) = ψI { coe := MagicFunction.Parametrisations.z₆' t, coe_im_pos := ⋯ }

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theorem ψS_slash_ST_apply (z : UpperHalfPlane) :

SlashAction.map (-2) (ModularGroup.S * ModularGroup.T) ψS z = ψS { coe := -1 / (↑z + 1), coe_im_pos := ⋯ } * (↑z + 1) ^ 2

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theorem ψS_slash_ST_apply' (z : UpperHalfPlane) :

SlashAction.map (-2) (ModularGroup.S * ModularGroup.T) ψS z = ψS' (-1 / (↑z + 1)) * (↑z + 1) ^ 2

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theorem ψS_slash_S_apply (z : UpperHalfPlane) :

SlashAction.map (-2) ModularGroup.S ψS z = ψS { coe := -1 / ↑z, coe_im_pos := ⋯ } * ↑z ^ 2

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theorem ψS_slash_S_apply' (z : UpperHalfPlane) :

SlashAction.map (-2) ModularGroup.S ψS z = ψS' (-1 / ↑z) * ↑z ^ 2

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theorem ψS_slash_ST_explicit₁ {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψT' (MagicFunction.Parametrisations.z₁' t) = ψS' (-1 / (MagicFunction.Parametrisations.z₁' t + 1)) * (MagicFunction.Parametrisations.z₁' t + 1) ^ 2

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theorem ψS_slash_ST_explicit₂ {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψT' (MagicFunction.Parametrisations.z₂' t) = ψS' (-1 / (MagicFunction.Parametrisations.z₂' t + 1)) * (MagicFunction.Parametrisations.z₂' t + 1) ^ 2

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theorem ψS_slash_ST_explicit₃ {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψT' (MagicFunction.Parametrisations.z₃' t) = ψS' (-1 / (MagicFunction.Parametrisations.z₃' t + 1)) * (MagicFunction.Parametrisations.z₃' t + 1) ^ 2

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theorem ψS_slash_ST_explicit₄ {t : ℝ} (ht : t ∈ Set.Icc 0 1) :

ψT' (MagicFunction.Parametrisations.z₄' t) = ψS' (-1 / (MagicFunction.Parametrisations.z₄' t + 1)) * (MagicFunction.Parametrisations.z₄' t + 1) ^ 2

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theorem ψS_slash_S_explicit₅ {t : ℝ} (ht : t ∈ Set.Ioc 0 1) :

ψI' (MagicFunction.Parametrisations.z₅' t) = ψS' (-1 / MagicFunction.Parametrisations.z₅' t) * MagicFunction.Parametrisations.z₅' t ^ 2

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theorem ψS_slash_ST_explicit₆ {t : ℝ} (ht : t ∈ Set.Ici 1) :

ψT' (MagicFunction.Parametrisations.z₆' t) = ψS' (-1 / (MagicFunction.Parametrisations.z₆' t + 1)) * (MagicFunction.Parametrisations.z₆' t + 1) ^ 2

Documentation

SpherePacking.MagicFunction.b.psi

The ψ Functions #