# Modular lambda function

## Definitions

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

## Illustrations

Image: X-ray of $\lambda(\tau)$ on $\tau \in \left[-\frac{3}{2}, \frac{3}{2}\right] + \left[0, 2\right] i$ with $\mathcal{F}_{\lambda}$ highlighted
Image: Plot of $\lambda\!\left(i x\right)$ on $x \in \left[0, 4\right]$

## Domain

$\tau \in \mathbb{H} \;\implies\; \lambda(\tau) \in \mathbb{C} \setminus \left\{0, 1\right\}$
$\tau \in \left\{i \infty\right\} \;\implies\; \lambda(\tau) \in \left\{0\right\}$
$\tau \in \left\{ 2 n : n \in \mathbb{Z} \right\} \;\implies\; \lambda(\tau) \in \left\{1\right\}$
$\lambda(\tau) \text{ is holomorphic on } \tau \in \mathbb{H} \cup \left\{i \infty\right\}$

## Modular transformations

### Level 2 principal subgroup

$\lambda\!\left(\tau + 2\right) = \lambda(\tau)$
$\lambda\!\left(\frac{\tau}{2 \tau + 1}\right) = \lambda(\tau)$
$\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda(\tau)$

### Arbitrary modular transformations

$\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda(\tau), 1 - \lambda(\tau), \frac{1}{\lambda(\tau)}, \frac{1}{1 - \lambda(\tau)}, \frac{\lambda(\tau) - 1}{\lambda(\tau)}, \frac{\lambda(\tau)}{\lambda(\tau) - 1}\right\}$
$\lambda\!\left(\tau + 1\right) = \frac{\lambda(\tau)}{\lambda(\tau) - 1}$
$\lambda\!\left(-\frac{1}{\tau}\right) = 1 - \lambda(\tau)$
$\lambda\!\left(\frac{\tau}{1 - \tau}\right) = \frac{1}{\lambda(\tau)}$
$\lambda\!\left(\frac{1}{1 - \tau}\right) = \frac{1}{1 - \lambda(\tau)}$
$\lambda\!\left(\frac{\tau - 1}{\tau}\right) = \frac{\lambda(\tau) - 1}{\lambda(\tau)}$
$\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \begin{cases} \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(1, 0, 0, 1\right) \pmod {2}\\1 - \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 0\right) \pmod {2}\\\frac{1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 0, 1, 1\right) \pmod {2}\\\frac{1}{1 - \lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 1\right) \pmod {2}\\\frac{\lambda(\tau) - 1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 1, 1, 0\right) \pmod {2}\\\frac{\lambda(\tau)}{\lambda(\tau) - 1}, & \left(a, b, c, d\right) \equiv \left(1, 1, 0, 1\right) \pmod {2}\\ \end{cases}$

### Fundamental domain

$\mathcal{F}_{\lambda} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left(\left(\operatorname{Re}(\tau) \in \left(-1, 1\right) \;\mathbin{\operatorname{and}}\; \min\!\left(\left|\tau - \frac{1}{2}\right|, \left|z + \frac{1}{2}\right|\right) > \frac{1}{2}\right) \;\mathbin{\operatorname{or}}\; \operatorname{Re}(\tau) = -1 \;\mathbin{\operatorname{or}}\; \left|\tau + \frac{1}{2}\right| = \frac{1}{2}\right) \right\}$
$\mathbb{H} = \left\{ \gamma \circ \tau : \tau \in \mathcal{F}_{\lambda} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \gamma \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2} \right\}$

## Theta function representations

$\lambda(\tau) = \frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}$
$\frac{\lambda(\tau)}{\lambda(\tau) - 1} = -\frac{\theta_{2}^{4}\!\left(0, \tau\right)}{\theta_{4}^{4}\!\left(0, \tau\right)}$
$1 - \lambda(\tau) = \frac{\theta_{4}^{4}\!\left(0, \tau\right)}{\theta_{3}^{4}\!\left(0, \tau\right)}$

## Dedekind eta function representations

$\lambda(\tau) = 16 \frac{\eta^{8}\!\left(\frac{\tau}{2}\right) \eta^{16}\!\left(2 \tau\right)}{\eta^{24}\!\left(\tau\right)}$
$\frac{1}{\lambda(\tau)} = \frac{1}{16} \frac{\eta^{8}\!\left(\frac{\tau}{2}\right)}{\eta^{8}\!\left(2 \tau\right)} + 1$

## Elliptic function representations

$\lambda(\tau) = \frac{\wp\!\left(\frac{1}{2} \left(1 + \tau\right), \tau\right) - \wp\!\left(\frac{\tau}{2}, \tau\right)}{\wp\!\left(\frac{1}{2}, \tau\right) - \wp\!\left(\frac{\tau}{2}, \tau\right)}$

## Fourier series (q-series)

$\lambda(\tau) = 16 q - 128 {q}^{2} + 704 {q}^{3} - 3072 {q}^{4} + 11488 {q}^{5} - 38400 {q}^{6} + \ldots \; \text{ where } q = {e}^{\pi i \tau}$
$\lambda(\tau) = 16 q \prod_{k=1}^{\infty} {\left(\frac{1 + {q}^{2 k}}{1 + {q}^{2 k - 1}}\right)}^{8}\; \text{ where } q = {e}^{\pi i \tau}$
$a(n) \sim {\left(-1\right)}^{n + 1} \frac{{e}^{2 \pi \sqrt{n}}}{32 {n}^{3 / 4}}, \; n \to \infty\; \text{ where } a(n) = [q^{n}] \lambda(\tau) \; \left(q = {e}^{\pi i \tau}\right)$

## Range

$\left\{ \lambda(\tau) : \tau \in \mathbb{H} \right\} = \left\{ \lambda(\tau) : \tau \in \mathcal{F}_{\lambda} \right\} = \mathbb{C} \setminus \left\{0, 1\right\}$
$\left\{ \lambda(\tau) : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) = -1 \right\} = \left(-\infty, 0\right)$
$\left\{ \lambda(\tau) : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\tau + \frac{1}{2}\right| = \frac{1}{2} \right\} = \left(1, \infty\right)$
$\left\{ \lambda(\tau) : \tau \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \right\} = \mathbb{C} \setminus \left(\left(-\infty, 0\right] \cup \left[1, \infty\right)\right)$

## Specific values

$\lambda(i) = \frac{1}{2}$
$\lambda\!\left(1 + i\right) = -1$
$\lambda\!\left(\frac{1 + i}{2}\right) = 2$
$\lambda\!\left(\frac{a i + b}{c i + d}\right) \in \left\{-1, \frac{1}{2}, 2\right\}$
$\lambda(\omega) = -\omega\; \text{ where } \omega = {e}^{2 \pi i / 3}$
$\lambda\!\left(\frac{i}{2}\right) = 12 \sqrt{2} - 16$
$\lambda\!\left(2 i\right) = 17 - 12 \sqrt{2}$

### Limiting values

$\lambda\!\left(i \infty\right) = \lim_{\tau \to i \infty} \lambda(\tau) = 0$
$\lim_{\varepsilon \to {0}^{+}} \lambda\!\left(n + i \varepsilon\right) = \begin{cases} 1, & n \text{ even}\\-\infty, & n \text{ odd}\\ \end{cases}$

## Inverse and transcendental equations

$\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)}$
$\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil$

## Connection to the j-invariant

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

## Derivatives

$\lambda'(\tau) = \frac{\pi i}{3} \left(E_{2}\!\left(\frac{\tau}{2}\right) + 8 E_{2}\!\left(2 \tau\right) - 6 E_{2}\!\left(\tau\right)\right) \lambda(\tau)$
$\lambda'(\tau) = \frac{2 i}{\pi} \left(\zeta\!\left(\frac{1}{2}, \frac{\tau}{2}\right) + 8 \zeta\!\left(\frac{1}{2}, 2 \tau\right) - 6 \zeta\!\left(\frac{1}{2}, \tau\right)\right) \lambda(\tau)$
$\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda(\tau)\right)\right)}^{2} \left(\lambda(\tau) - 1\right) \lambda(\tau)$

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

2020-08-27 09:56:25.682319 UTC