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Modular lambda function

Table of contents: Definitions - Illustrations - Domain - Modular transformations - Theta function representations - Dedekind eta function representations - Elliptic function representations - Fourier series (q-series) - Range - Specific values - Inverse and transcendental equations - Connection to the j-invariant - Derivatives

Definitions

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Symbol: ModularLambda — λ(τ)\lambda(\tau) — Modular lambda function
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Symbol: ModularLambdaFundamentalDomain — Fλ\mathcal{F}_{\lambda} — Fundamental domain of the modular lambda function

Illustrations

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Image: X-ray of λ(τ)\lambda(\tau) on τ∈[−32,32]+[0,2]i\tau \in \left[-\frac{3}{2}, \frac{3}{2}\right] + \left[0, 2\right] i with Fλ\mathcal{F}_{\lambda} highlighted
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Image: Plot of λ ⁣(ix)\lambda\!\left(i x\right) on x∈[0,4]x \in \left[0, 4\right]

Domain

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τ∈H    ⟹    λ(τ)∈C∖{0,1}\tau \in \mathbb{H} \;\implies\; \lambda(\tau) \in \mathbb{C} \setminus \left\{0, 1\right\}
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τ∈{i∞}    ⟹    λ(τ)∈{0}\tau \in \left\{i \infty\right\} \;\implies\; \lambda(\tau) \in \left\{0\right\}
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τ∈{2n:n∈Z}    ⟹    λ(τ)∈{1}\tau \in \left\{ 2 n : n \in \mathbb{Z} \right\} \;\implies\; \lambda(\tau) \in \left\{1\right\}
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λ(τ) is holomorphic on τ∈H∪{i∞}\lambda(\tau) \text{ is holomorphic on } \tau \in \mathbb{H} \cup \left\{i \infty\right\}

Modular transformations

Level 2 principal subgroup

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λ ⁣(τ+2)=λ(τ)\lambda\!\left(\tau + 2\right) = \lambda(\tau)
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λ ⁣(τ2τ+1)=λ(τ)\lambda\!\left(\frac{\tau}{2 \tau + 1}\right) = \lambda(\tau)
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λ ⁣(aτ+bcτ+d)=λ(τ)\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda(\tau)

Arbitrary modular transformations

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λ ⁣(aτ+bcτ+d)∈{λ(τ),1−λ(τ),1λ(τ),11−λ(τ),λ(τ)−1λ(τ),λ(τ)λ(τ)−1}\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda(\tau), 1 - \lambda(\tau), \frac{1}{\lambda(\tau)}, \frac{1}{1 - \lambda(\tau)}, \frac{\lambda(\tau) - 1}{\lambda(\tau)}, \frac{\lambda(\tau)}{\lambda(\tau) - 1}\right\}
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λ ⁣(τ+1)=λ(τ)λ(τ)−1\lambda\!\left(\tau + 1\right) = \frac{\lambda(\tau)}{\lambda(\tau) - 1}
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λ ⁣(−1τ)=1−λ(τ)\lambda\!\left(-\frac{1}{\tau}\right) = 1 - \lambda(\tau)
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λ ⁣(τ1−τ)=1λ(τ)\lambda\!\left(\frac{\tau}{1 - \tau}\right) = \frac{1}{\lambda(\tau)}
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λ ⁣(11−τ)=11−λ(τ)\lambda\!\left(\frac{1}{1 - \tau}\right) = \frac{1}{1 - \lambda(\tau)}
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λ ⁣(τ−1τ)=λ(τ)−1λ(τ)\lambda\!\left(\frac{\tau - 1}{\tau}\right) = \frac{\lambda(\tau) - 1}{\lambda(\tau)}
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λ ⁣(aτ+bcτ+d)={λ(τ),(a,b,c,d)≡(1,0,0,1)(mod2)1−λ(τ),(a,b,c,d)≡(0,1,1,0)(mod2)1λ(τ),(a,b,c,d)≡(1,0,1,1)(mod2)11−λ(τ),(a,b,c,d)≡(0,1,1,1)(mod2)λ(τ)−1λ(τ),(a,b,c,d)≡(1,1,1,0)(mod2)λ(τ)λ(τ)−1,(a,b,c,d)≡(1,1,0,1)(mod2)\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \begin{cases} \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(1, 0, 0, 1\right) \pmod {2}\\1 - \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 0\right) \pmod {2}\\\frac{1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 0, 1, 1\right) \pmod {2}\\\frac{1}{1 - \lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 1\right) \pmod {2}\\\frac{\lambda(\tau) - 1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 1, 1, 0\right) \pmod {2}\\\frac{\lambda(\tau)}{\lambda(\tau) - 1}, & \left(a, b, c, d\right) \equiv \left(1, 1, 0, 1\right) \pmod {2}\\ \end{cases}

Fundamental domain

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Fλ={τ:τ∈H  and⁡  ((Re⁡(τ)∈(−1,1)  and⁡  min⁡ ⁣(∣τ−12∣,∣z+12∣)>12)  or⁡  Re⁡(τ)=−1  or⁡  ∣τ+12∣=12)}\mathcal{F}_{\lambda} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left(\left(\operatorname{Re}(\tau) \in \left(-1, 1\right) \;\mathbin{\operatorname{and}}\; \min\!\left(\left|\tau - \frac{1}{2}\right|, \left|z + \frac{1}{2}\right|\right) > \frac{1}{2}\right) \;\mathbin{\operatorname{or}}\; \operatorname{Re}(\tau) = -1 \;\mathbin{\operatorname{or}}\; \left|\tau + \frac{1}{2}\right| = \frac{1}{2}\right) \right\}
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H={γ∘τ:τ∈Fλ  and⁡  γ∈SL⁡2(Z)  and⁡  γ≡(1001)(mod2)}\mathbb{H} = \left\{ \gamma \circ \tau : \tau \in \mathcal{F}_{\lambda} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \gamma \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2} \right\}

Theta function representations

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λ(τ)=θ24 ⁣(0,τ)θ34 ⁣(0,τ)\lambda(\tau) = \frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}
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λ(τ)λ(τ)−1=−θ24 ⁣(0,τ)θ44 ⁣(0,τ)\frac{\lambda(\tau)}{\lambda(\tau) - 1} = -\frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{4}^{4}\!\left(0, \tau\right)}
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1−λ(τ)=θ44 ⁣(0,τ)θ34 ⁣(0,τ)1 - \lambda(\tau) = \frac{\theta_{4}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}

Dedekind eta function representations

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λ(τ)=16η8 ⁣(τ2)η16 ⁣(2τ)η24 ⁣(τ)\lambda(\tau) = 16 \frac{\eta^{8}\!\left(\frac{\tau}{2}\right) \eta^{16}\!\left(2 \tau\right)}{\eta^{24}\!\left(\tau\right)}
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1λ(τ)=116η8 ⁣(τ2)η8 ⁣(2τ)+1\frac{1}{\lambda(\tau)} = \frac{1}{16} \frac{\eta^{8}\!\left(\frac{\tau}{2}\right)}{\eta^{8}\!\left(2 \tau\right)} + 1

Elliptic function representations

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λ(τ)=℘ ⁣(12(1+τ),τ)−℘ ⁣(τ2,τ)℘ ⁣(12,τ)−℘ ⁣(τ2,τ)\lambda(\tau) = \frac{\wp\!\left(\frac{1}{2} \left(1 + \tau\right), \tau\right) - \wp\!\left(\frac{\tau}{2}, \tau\right)}{\wp\!\left(\frac{1}{2}, \tau\right) - \wp\!\left(\frac{\tau}{2}, \tau\right)}

Fourier series (q-series)

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λ(τ)=16q−128q2+704q3−3072q4+11488q5−38400q6+…   where q=eπiτ\lambda(\tau) = 16 q - 128 {q}^{2} + 704 {q}^{3} - 3072 {q}^{4} + 11488 {q}^{5} - 38400 {q}^{6} + \ldots \; \text{ where } q = {e}^{\pi i \tau}
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λ(τ)=16q∏k=1∞(1+q2k1+q2k−1)8   where q=eπiτ\lambda(\tau) = 16 q \prod_{k=1}^{\infty} {\left(\frac{1 + {q}^{2 k}}{1 + {q}^{2 k - 1}}\right)}^{8}\; \text{ where } q = {e}^{\pi i \tau}
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a(n)∼(−1)n+1e2πn32n3/4,  n→∞   where a(n)=[qn]λ(τ)  (q=eπiτ)a(n) \sim {\left(-1\right)}^{n + 1} \frac{{e}^{2 \pi \sqrt{n}}}{32 {n}^{3 / 4}}, \; n \to \infty\; \text{ where } a(n) = [q^{n}] \lambda(\tau) \; \left(q = {e}^{\pi i \tau}\right)

Range

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{λ(τ):τ∈H}={λ(τ):τ∈Fλ}=C∖{0,1}\left\{ \lambda(\tau) : \tau \in \mathbb{H} \right\} = \left\{ \lambda(\tau) : \tau \in \mathcal{F}_{\lambda} \right\} = \mathbb{C} \setminus \left\{0, 1\right\}
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{λ(τ):τ∈H and⁡ Re⁡(τ)=−1}=(−∞,0)\left\{ \lambda(\tau) : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(\tau) = -1 \right\} = \left(-\infty, 0\right)
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{λ(τ):τ∈H and⁡ ∣τ+12∣=12}=(1,∞)\left\{ \lambda(\tau) : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left|\tau + \frac{1}{2}\right| = \frac{1}{2} \right\} = \left(1, \infty\right)
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{λ(τ):τ∈Interior⁡(Fλ)}=C∖((−∞,0]∪[1,∞))\left\{ \lambda(\tau) : \tau \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \right\} = \mathbb{C} \setminus \left(\left(-\infty, 0\right] \cup \left[1, \infty\right)\right)

Specific values

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λ(i)=12\lambda(i) = \frac{1}{2}
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λ ⁣(1+i)=−1\lambda\!\left(1 + i\right) = -1
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λ ⁣(1+i2)=2\lambda\!\left(\frac{1 + i}{2}\right) = 2
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λ ⁣(ai+bci+d)∈{−1,12,2}\lambda\!\left(\frac{a i + b}{c i + d}\right) \in \left\{-1, \frac{1}{2}, 2\right\}
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λ(ω)=−ω   where ω=e2πi/3\lambda(\omega) = -\omega\; \text{ where } \omega = {e}^{2 \pi i / 3}
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λ ⁣(i2)=122−16\lambda\!\left(\frac{i}{2}\right) = 12 \sqrt{2} - 16
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λ ⁣(2i)=17−122\lambda\!\left(2 i\right) = 17 - 12 \sqrt{2}

Limiting values

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λ ⁣(i∞)=lim⁡τ→i∞λ(τ)=0\lambda\!\left(i \infty\right) = \lim_{\tau \to i \infty} \lambda(\tau) = 0
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lim⁡ε→0+λ ⁣(n+iε)={1,n even−∞,n odd\lim_{\varepsilon \to {0}^{+}} \lambda\!\left(n + i \varepsilon\right) = \begin{cases} 1, & n \text{ even}\\-\infty, & n \text{ odd}\\ \end{cases}

Inverse and transcendental equations

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τ=iK ⁣(1−λ(τ))K ⁣(λ(τ))\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)}
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τ=iK ⁣(1−λ(τ))K ⁣(λ(τ))+2⌈12Re⁡(τ)−12⌉\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil

Connection to the j-invariant

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j(τ)=256(1−λ(τ)+(λ(τ))2)3(λ(τ))2(1−λ(τ))2j(\tau) = 256 \frac{{\left(1 - \lambda(\tau) + {\left(\lambda(\tau)\right)}^{2}\right)}^{3}}{{\left(\lambda(\tau)\right)}^{2} {\left(1 - \lambda(\tau)\right)}^{2}}

Derivatives

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λ′(τ)=πi3(E2 ⁣(τ2)+8E2 ⁣(2τ)−6E2 ⁣(τ))λ(τ)\lambda'(\tau) = \frac{\pi i}{3} \left(E_{2}\!\left(\frac{\tau}{2}\right) + 8 E_{2}\!\left(2 \tau\right) - 6 E_{2}\!\left(\tau\right)\right) \lambda(\tau)
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λ′(τ)=2iπ(ζ ⁣(12,τ2)+8ζ ⁣(12,2τ)−6ζ ⁣(12,τ))λ(τ)\lambda'(\tau) = \frac{2 i}{\pi} \left(\zeta\!\left(\frac{1}{2}, \frac{\tau}{2}\right) + 8 \zeta\!\left(\frac{1}{2}, 2 \tau\right) - 6 \zeta\!\left(\frac{1}{2}, \tau\right)\right) \lambda(\tau)
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λ′(τ)=−4iπ(K ⁣(λ(τ)))2(λ(τ)−1)λ(τ)\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda(\tau)\right)\right)}^{2} \left(\lambda(\tau) - 1\right) \lambda(\tau)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC