Modular lambda function

In mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence subgroup ⁠ $Γ (2)$ ⁠ of ${SL}_{2} (ℤ)$ , and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve ⁠ $X (2)$ ⁠. Over any point ⁠ $τ$ ⁠, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $ℂ / ⟨ 1, τ ⟩$ , where the map is defined as the quotient by the [−1] involution.

The q-expansion, where $q = e^{π i τ}$ is the nome, is given by

λ (τ) = 16 q - 128 q^{2} + 704 q^{3} - 3072 q^{4} + 11488 q^{5} - 38400 q^{6} + \dots

(sequence A115977 in the OEIS). By symmetrizing the lambda function under the canonical action of the symmetric group ⁠ $X (2)$ ⁠ on ⁠ $S_{3}$ ⁠, and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group ${SL}_{2} (ℤ)$ , and it is in fact Klein's modular j-invariant.

Modular properties

The function $λ (τ)$ is invariant under the group generated by

τ \mapsto τ + 2, τ \mapsto \frac{τ}{1 - 2 τ} .

The generators of the modular group act by

τ \mapsto τ + 1 : λ \mapsto \frac{λ}{λ - 1};

τ \mapsto - \frac{1}{τ} : λ \mapsto 1 - λ .

Consequently, the action of the modular group on $λ (τ)$ is that of the anharmonic group, giving the six values of the cross-ratio:

{λ, \frac{1}{1 - λ}, \frac{λ - 1}{λ}, \frac{1}{λ}, \frac{λ}{λ - 1}, 1 - λ} .

Relations to other functions

It is the square of the elliptic modulus, that is, $λ (τ) = k^{2} (τ)$ . In terms of the Dedekind eta function $η (τ)$ and theta functions,

λ (τ) = (\frac{\sqrt{2} η (\frac{τ}{2}) η^{2} (2 τ)}{η^{3} (τ)})^{8} = \frac{16}{{(\frac{η (τ / 2)}{η (2 τ)})}^{8} + 16} = \frac{θ_{2}^{4} (τ)}{θ_{3}^{4} (τ)}

and,

\frac{1}{(λ (τ))^{1 / 4}} - (λ (τ))^{1 / 4} = \frac{1}{2} {(\frac{η (\frac{τ}{4})}{η (τ)})}^{4} = 2 \frac{θ_{4}^{2} (\frac{τ}{2})}{θ_{2}^{2} (\frac{τ}{2})}

where

θ_{2} (τ) = \sum_{n = - \infty}^{\infty} e^{π i τ (n + 1 / 2)^{2}}

θ_{3} (τ) = \sum_{n = - \infty}^{\infty} e^{π i τ n^{2}}

θ_{4} (τ) = \sum_{n = - \infty}^{\infty} (- 1)^{n} e^{π i τ n^{2}}

In terms of the half-periods of Weierstrass's elliptic functions, let $[ω_{1}, ω_{2}]$ be a fundamental pair of periods with $τ = \frac{ω_{2}}{ω_{1}}$ .

e_{1} = ℘ (\frac{ω_{1}}{2}), e_{2} = ℘ (\frac{ω_{2}}{2}), e_{3} = ℘ (\frac{ω_{1} + ω_{2}}{2})

we have

λ = \frac{e_{3} - e_{2}}{e_{1} - e_{2}} .

Since the three half-period values are distinct, this shows that $λ$ does not take the value 0 or 1.

The relation to the j-invariant is

j (τ) = \frac{256 (1 - λ (1 - λ))^{3}}{(λ (1 - λ))^{2}} = \frac{256 (1 - λ + λ^{2})^{3}}{λ^{2} (1 - λ)^{2}} .

which is the j-invariant of the elliptic curve of Legendre form $y^{2} = x (x - 1) (x - λ)$

Given $m \in ℂ ∖ {0, 1}$ , let

τ = i \frac{K {1 - m}}{K {m}}

where $K$ is the complete elliptic integral of the first kind with parameter $m = k^{2}$ . Then

λ (τ) = m .

Modular equations

The modular equation of degree $p$ (where $p$ is a prime number) is an algebraic equation in $λ (p τ)$ and $λ (τ)$ . If $λ (p τ) = u^{8}$ and $λ (τ) = v^{8}$ , the modular equations of degrees $p = 2, 3, 5, 7$ are, respectively,

(1 + u^{4})^{2} v^{8} - 4 u^{4} = 0,

u^{4} - v^{4} + 2 u v (1 - u^{2} v^{2}) = 0,

u^{6} - v^{6} + 5 u^{2} v^{2} (u^{2} - v^{2}) + 4 u v (1 - u^{4} v^{4}) = 0,

(1 - u^{8}) (1 - v^{8}) - (1 - u v)^{8} = 0.

The quantity $v$ (and hence $u$ ) can be thought of as a holomorphic function on the upper half-plane $Im τ > 0$ :

\begin{matrix} v & = \prod_{k = 1}^{\infty} \tanh \frac{(k - 1 / 2) π i}{τ} = \sqrt{2} e^{π i τ / 8} \frac{\sum_{k \in ℤ} e^{(2 k^{2} + k) π i τ}}{\sum_{k \in ℤ} e^{k^{2} π i τ}} \\ = \frac{\sqrt{2} e^{π i τ / 8}}{1 + \frac{e^{π i τ}}{1 + e^{π i τ} + \frac{e^{2 π i τ}}{1 + e^{2 π i τ} + \frac{e^{3 π i τ}}{1 + e^{3 π i τ} + ⋱}}}} \end{matrix}

Since $λ (i) = 1 / 2$ , the modular equations can be used to give algebraic values of $λ (p i)$ for any prime $p$ . The algebraic values of $λ (n i)$ are also given by

λ (n i) = \prod_{k = 1}^{n / 2} {sl}^{8} \frac{(2 k - 1) ϖ}{2 n} (n even)

λ (n i) = \frac{1}{2^{n}} \prod_{k = 1}^{n - 1} {(1 - {sl}^{2} \frac{k ϖ}{n})}^{2} (n odd)

where $sl$ is the lemniscate sine and $ϖ$ is the lemniscate constant.

Lambda-star

Definition and computation of lambda-star

The function $λ^{*} (x)$ (where $x \in ℝ^{+}$ ) gives the value of the elliptic modulus $k$ , for which the complete elliptic integral of the first kind $K (k)$ and its complementary counterpart $K (\sqrt{1 - k^{2}})$ are related by following expression:

\frac{K [\sqrt{1 - λ^{*} (x)^{2}}]}{K [λ^{*} (x)]} = \sqrt{x}

The values of $λ^{*} (x)$ can be computed as follows:

λ^{*} (x) = \frac{θ_{2}^{2} (i \sqrt{x})}{θ_{3}^{2} (i \sqrt{x})}

λ^{*} (x) = {[\sum_{a = - \infty}^{\infty} \exp [- (a + 1 / 2)^{2} π \sqrt{x}]]}^{2} {[\sum_{a = - \infty}^{\infty} \exp (- a^{2} π \sqrt{x})]}^{- 2}

λ^{*} (x) = [\sum_{a = - \infty}^{\infty} sech [(a + 1 / 2) π \sqrt{x}]] {[\sum_{a = - \infty}^{\infty} sech (a π \sqrt{x})]}^{- 1}

The functions $λ^{*}$ and $λ$ are related to each other in this way:

λ^{*} (x) = \sqrt{λ (i \sqrt{x})}

Properties of lambda-star

Every $λ^{*}$ value of a positive rational number is a positive algebraic number:

λ^{*} (x) \in {\overline{ℚ}}_{+} \forall x \in ℚ^{+} .

$K (λ^{*} (x))$ and $E (λ^{*} (x))$ (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any $x \in ℚ^{+}$ , as Selberg and Chowla proved in 1949.

The following expression is valid for all $n \in ℕ$ :

\sqrt{n} = \sum_{a = 1}^{n} dn [\frac{2 a}{n} K [λ^{*} (\frac{1}{n})]; λ^{*} (\frac{1}{n})]

where $dn$ is the Jacobi elliptic function delta amplitudinis with modulus $k$ .

By knowing one $λ^{*}$ value, this formula can be used to compute related $λ^{*}$ values:

λ^{*} (n^{2} x) = λ^{*} (x)^{n} \prod_{a = 1}^{n} sn {\frac{2 a - 1}{n} K [λ^{*} (x)]; λ^{*} (x)}^{2}

where $n \in ℕ$ and $sn$ is the Jacobi elliptic function sinus amplitudinis with modulus $k$ .

Further relations:

λ^{*} (x)^{2} + λ^{*} (1 / x)^{2} = 1

[λ^{*} (x) + 1] [λ^{*} (4 / x) + 1] = 2

λ^{*} (4 x) = \frac{1 - \sqrt{1 - λ^{*} (x)^{2}}}{1 + \sqrt{1 - λ^{*} (x)^{2}}} = \tan {\frac{1}{2} \arcsin [λ^{*} (x)]}^{2}

λ^{*} (x) - λ^{*} (9 x) = 2 [λ^{*} (x) λ^{*} (9 x)]^{1 / 4} - 2 [λ^{*} (x) λ^{*} (9 x)]^{3 / 4}

$\begin{matrix} a^{6} - f^{6} = 2 a f + 2 a^{5} f^{5} & (a = {[\frac{2 λ^{*} (x)}{1 - λ^{*} (x)^{2}}]}^{1 / 12}) & (f = {[\frac{2 λ^{*} (25 x)}{1 - λ^{*} (25 x)^{2}}]}^{1 / 12}) \\ a^{8} + b^{8} - 7 a^{4} b^{4} = 2 \sqrt{2} a b + 2 \sqrt{2} a^{7} b^{7} & (a = {[\frac{2 λ^{*} (x)}{1 - λ^{*} (x)^{2}}]}^{1 / 12}) & (b = {[\frac{2 λ^{*} (49 x)}{1 - λ^{*} (49 x)^{2}}]}^{1 / 12}) \\ a^{12} - c^{12} = 2 \sqrt{2} (a c + a^{3} c^{3}) (1 + 3 a^{2} c^{2} + a^{4} c^{4}) (2 + 3 a^{2} c^{2} + 2 a^{4} c^{4}) & (a = {[\frac{2 λ^{*} (x)}{1 - λ^{*} (x)^{2}}]}^{1 / 12}) & (c = {[\frac{2 λ^{*} (121 x)}{1 - λ^{*} (121 x)^{2}}]}^{1 / 12}) \\ (a^{2} - d^{2}) (a^{4} + d^{4} - 7 a^{2} d^{2}) [(a^{2} - d^{2})^{4} - a^{2} d^{2} (a^{2} + d^{2})^{2}] = 8 a d + 8 a^{13} d^{13} & (a = {[\frac{2 λ^{*} (x)}{1 - λ^{*} (x)^{2}}]}^{1 / 12}) & (d = {[\frac{2 λ^{*} (169 x)}{1 - λ^{*} (169 x)^{2}}]}^{1 / 12}) \end{matrix}$

Special values

Lambda-star values of integer numbers of 4n-3-type:

λ^{*} (1) = \frac{1}{\sqrt{2}}

λ^{*} (5) = \sin [\frac{1}{2} \arcsin (\sqrt{5} - 2)]

λ^{*} (9) = \frac{1}{2} (\sqrt{3} - 1) (\sqrt{2} - \sqrt[4]{3})

λ^{*} (13) = \sin [\frac{1}{2} \arcsin (5 \sqrt{13} - 18)]

λ^{*} (17) = \sin {\frac{1}{2} \arcsin [\frac{1}{64} {(5 + \sqrt{17} - \sqrt{10 \sqrt{17} + 26})}^{3}]}

λ^{*} (21) = \sin {\frac{1}{2} \arcsin [(8 - 3 \sqrt{7}) (2 \sqrt{7} - 3 \sqrt{3})]}

λ^{*} (25) = \frac{1}{\sqrt{2}} (\sqrt{5} - 2) (3 - 2 \sqrt[4]{5})

λ^{*} (33) = \sin {\frac{1}{2} \arcsin [(10 - 3 \sqrt{11}) (2 - \sqrt{3})^{3}]}

λ^{*} (37) = \sin {\frac{1}{2} \arcsin [(\sqrt{37} - 6)^{3}]}

λ^{*} (45) = \sin {\frac{1}{2} \arcsin [(4 - \sqrt{15})^{2} (\sqrt{5} - 2)^{3}]}

λ^{*} (49) = \frac{1}{4} (8 + 3 \sqrt{7}) (5 - \sqrt{7} - \sqrt[4]{28}) (\sqrt{14} - \sqrt{2} - \sqrt[8]{28} \sqrt{5 - \sqrt{7}})

λ^{*} (57) = \sin {\frac{1}{2} \arcsin [(170 - 39 \sqrt{19}) (2 - \sqrt{3})^{3}]}

λ^{*} (73) = \sin {\frac{1}{2} \arcsin [\frac{1}{64} {(45 + 5 \sqrt{73} - 3 \sqrt{50 \sqrt{73} + 426})}^{3}]}

Lambda-star values of integer numbers of 4n-2-type:

λ^{*} (2) = \sqrt{2} - 1

λ^{*} (6) = (2 - \sqrt{3}) (\sqrt{3} - \sqrt{2})

λ^{*} (10) = (\sqrt{10} - 3) (\sqrt{2} - 1)^{2}

λ^{*} (14) = \tan {\frac{1}{2} \arctan [\frac{1}{8} {(2 \sqrt{2} + 1 - \sqrt{4 \sqrt{2} + 5})}^{3}]}

λ^{*} (18) = (\sqrt{2} - 1)^{3} (2 - \sqrt{3})^{2}

λ^{*} (22) = (10 - 3 \sqrt{11}) (3 \sqrt{11} - 7 \sqrt{2})

λ^{*} (30) = \tan {\frac{1}{2} \arctan [(\sqrt{10} - 3)^{2} (\sqrt{5} - 2)^{2}]}

λ^{*} (34) = \tan {\frac{1}{4} \arcsin [\frac{1}{9} (\sqrt{17} - 4)^{2}]}

λ^{*} (42) = \tan {\frac{1}{2} \arctan [(2 \sqrt{7} - 3 \sqrt{3})^{2} (2 \sqrt{2} - \sqrt{7})^{2}]}

λ^{*} (46) = \tan {\frac{1}{2} \arctan [\frac{1}{64} {(3 + \sqrt{2} - \sqrt{6 \sqrt{2} + 7})}^{6}]}

λ^{*} (58) = (13 \sqrt{58} - 99) (\sqrt{2} - 1)^{6}

λ^{*} (70) = \tan {\frac{1}{2} \arctan [(\sqrt{5} - 2)^{4} (\sqrt{2} - 1)^{6}]}

λ^{*} (78) = \tan {\frac{1}{2} \arctan [(5 \sqrt{13} - 18)^{2} (\sqrt{26} - 5)^{2}]}

λ^{*} (82) = \tan {\frac{1}{4} \arcsin [\frac{1}{4761} (8 \sqrt{41} - 51)^{2}]}

Lambda-star values of integer numbers of 4n-1-type:

λ^{*} (3) = \frac{1}{2 \sqrt{2}} (\sqrt{3} - 1)

λ^{*} (7) = \frac{1}{4 \sqrt{2}} (3 - \sqrt{7})

λ^{*} (11) = \frac{1}{8 \sqrt{2}} (\sqrt{11} + 3) {(\frac{1}{3} \sqrt[3]{6 \sqrt{3} + 2 \sqrt{11}} - \frac{1}{3} \sqrt[3]{6 \sqrt{3} - 2 \sqrt{11}} + \frac{1}{3} \sqrt{11} - 1)}^{4}

λ^{*} (15) = \frac{1}{8 \sqrt{2}} (3 - \sqrt{5}) (\sqrt{5} - \sqrt{3}) (2 - \sqrt{3})

λ^{*} (19) = \frac{1}{8 \sqrt{2}} (3 \sqrt{19} + 13) {[\frac{1}{6} (\sqrt{19} - 2 + \sqrt{3}) \sqrt[3]{3 \sqrt{3} - \sqrt{19}} - \frac{1}{6} (\sqrt{19} - 2 - \sqrt{3}) \sqrt[3]{3 \sqrt{3} + \sqrt{19}} - \frac{1}{3} (5 - \sqrt{19})]}^{4}

λ^{*} (23) = \frac{1}{16 \sqrt{2}} (5 + \sqrt{23}) {[\frac{1}{6} (\sqrt{3} + 1) \sqrt[3]{100 - 12 \sqrt{69}} - \frac{1}{6} (\sqrt{3} - 1) \sqrt[3]{100 + 12 \sqrt{69}} + \frac{2}{3}]}^{4}

λ^{*} (27) = \frac{1}{16 \sqrt{2}} (\sqrt{3} - 1)^{3} {[\frac{1}{3} \sqrt{3} (\sqrt[3]{4} - \sqrt[3]{2} + 1) - \sqrt[3]{2} + 1]}^{4}

λ^{*} (39) = \sin {\frac{1}{2} \arcsin [\frac{1}{16} (6 - \sqrt{13} - 3 \sqrt{6 \sqrt{13} - 21})]}

λ^{*} (55) = \sin {\frac{1}{2} \arcsin [\frac{1}{512} {(3 \sqrt{5} - 3 - \sqrt{6 \sqrt{5} - 2})}^{3}]}

Lambda-star values of integer numbers of 4n-type:

λ^{*} (4) = (\sqrt{2} - 1)^{2}

λ^{*} (8) = {(\sqrt{2} + 1 - \sqrt{2 \sqrt{2} + 2})}^{2}

λ^{*} (12) = (\sqrt{3} - \sqrt{2})^{2} (\sqrt{2} - 1)^{2}

λ^{*} (16) = (\sqrt{2} + 1)^{2} (\sqrt[4]{2} - 1)^{4}

λ^{*} (20) = \tan {[\frac{1}{4} \arcsin (\sqrt{5} - 2)]}^{2}

λ^{*} (24) = \tan {\frac{1}{2} \arcsin [(2 - \sqrt{3}) (\sqrt{3} - \sqrt{2})]}^{2}

λ^{*} (28) = (2 \sqrt{2} - \sqrt{7})^{2} (\sqrt{2} - 1)^{4}

λ^{*} (32) = \tan {\frac{1}{2} \arcsin [{(\sqrt{2} + 1 - \sqrt{2 \sqrt{2} + 2})}^{2}]}^{2}

Lambda-star values of rational fractions:

λ^{*} (\frac{1}{2}) = \sqrt{2 \sqrt{2} - 2}

λ^{*} (\frac{1}{3}) = \frac{1}{2 \sqrt{2}} (\sqrt{3} + 1)

λ^{*} (\frac{2}{3}) = (2 - \sqrt{3}) (\sqrt{3} + \sqrt{2})

λ^{*} (\frac{1}{4}) = 2 \sqrt[4]{2} (\sqrt{2} - 1)

λ^{*} (\frac{3}{4}) = \sqrt[4]{8} (\sqrt{3} - \sqrt{2}) (\sqrt{2} + 1) \sqrt{(\sqrt{3} - 1)^{3}}

λ^{*} (\frac{1}{5}) = \frac{1}{2 \sqrt{2}} (\sqrt{2 \sqrt{5} - 2} + \sqrt{5} - 1)

λ^{*} (\frac{2}{5}) = (\sqrt{10} - 3) (\sqrt{2} + 1)^{2}

λ^{*} (\frac{3}{5}) = \frac{1}{8 \sqrt{2}} (3 + \sqrt{5}) (\sqrt{5} - \sqrt{3}) (2 + \sqrt{3})

λ^{*} (\frac{4}{5}) = \tan {[\frac{π}{4} - \frac{1}{4} \arcsin (\sqrt{5} - 2)]}^{2}

Ramanujan's class invariants

Ramanujan's class invariants $G_{n}$ and $g_{n}$ are defined as

G_{n} = 2^{- 1 / 4} e^{π \sqrt{n} / 24} \prod_{k = 0}^{\infty} (1 + e^{- (2 k + 1) π \sqrt{n}}),

g_{n} = 2^{- 1 / 4} e^{π \sqrt{n} / 24} \prod_{k = 0}^{\infty} (1 - e^{- (2 k + 1) π \sqrt{n}}),

where $n \in ℚ^{+}$ . For such $n$ , the class invariants are algebraic numbers. For example

g_{58} = \sqrt{\frac{5 + \sqrt{29}}{2}}, g_{190} = \sqrt{(\sqrt{5} + 2) (\sqrt{10} + 3)} .

Identities with the class invariants include

G_{n} = G_{1 / n}, g_{n} = \frac{1}{g_{4 / n}}, g_{4 n} = 2^{1 / 4} g_{n} G_{n} .

The class invariants are very closely related to the Weber modular functions $𝔣$ and $𝔣_{1}$ . These are the relations between lambda-star and the class invariants:

G_{n} = \sin {2 \arcsin [λ^{*} (n)]}^{- 1 / 12} = 1 / [\sqrt[12]{2 λ^{*} (n)} \sqrt[24]{1 - λ^{*} (n)^{2}}]

g_{n} = \tan {2 \arctan [λ^{*} (n)]}^{- 1 / 12} = \sqrt[12]{[1 - λ^{*} (n)^{2}] / [2 λ^{*} (n)]}

λ^{*} (n) = \tan {\frac{1}{2} \arctan [g_{n}^{- 12}]} = \sqrt{g_{n}^{24} + 1} - g_{n}^{12}

Other appearances

Little Picard theorem

The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879. Suppose if possible that $f$ is entire and does not take the values 0 and 1. Since $λ$ is holomorphic, it has a local holomorphic inverse $ω$ defined away from $0, 1, \infty$ . Consider the function $z \mapsto ω (f (z))$ . By the Monodromy theorem this is holomorphic and maps the complex plane $ℂ$ to the upper half plane. From this it is easy to construct a holomorphic function from $ℂ$ to the unit disc, which by Liouville's theorem must be constant.

Moonshine

The function $τ \mapsto 16 / λ (2 τ) - 8$ is the normalized Hauptmodul for the group $Γ_{0} (4)$ , and its q-expansion

q^{- 1} + 20 q - 62 q^{3} + \dots

(sequence A007248 in the OEIS), where $q = e^{2 π i τ}$ , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

Footnotes

↑ Chandrasekharan (1985), p. 115.
↑ Chandrasekharan (1985), p. 109.
↑ Chandrasekharan (1985), p. 110.
1 2 3 4 Chandrasekharan (1985), p. 108.
↑ Chandrasekharan (1985), p. 63.
↑ Chandrasekharan (1985), p. 117.
↑ Rankin (1977), pp. 226–228.
↑ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 103–109, 134
1 2 Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
↑ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 152
↑ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. doi:10.1073/PNAS.35.7.371. PMC 1063041. S2CID 45071481.
↑ Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110.
↑ Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". Transactions of the American Mathematical Society. 349 (6): 2125–2173.
↑ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240
↑ Chandrasekharan (1985), p. 121.
↑ Chandrasekharan (1985), p. 118.

References

Notes

↑ $λ (τ)$ is not a modular function (per the Wikipedia definition), but every modular function is a rational function in $λ (τ)$ . Some authors use a non-equivalent definition of "modular functions".
↑ For any prime power, we can iterate the modular equation of degree $p$ . This process can be used to give algebraic values of $λ (n i)$ for any $n \in ℕ .$
↑ $sl a ϖ$ is algebraic for every $a \in ℚ .$

Other

Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0, Zbl 0543.33001
Chandrasekharan, K. (1985), Elliptic Functions, Grundlehren der mathematischen Wissenschaften, vol. 281, Springer-Verlag, pp. 108–121, ISBN 3-540-15295-4, Zbl 0575.33001
Conway, John Horton; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society, 11 (3): 308–339, doi:10.1112/blms/11.3.308, MR 0554399, Zbl 0424.20010
Rankin, Robert A. (1977), Modular Forms and Functions, Cambridge University Press, ISBN 0-521-21212-X, Zbl 0376.10020
Reinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.

Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.

External links

Modular lambda function at Fungrim

[FOOTNOTEChandrasekharan1985115-2] Chandrasekharan (1985), p. 115.

[FOOTNOTEChandrasekharan1985109-3] Chandrasekharan (1985), p. 109.

[FOOTNOTEChandrasekharan1985110-4] Chandrasekharan (1985), p. 110.

[FOOTNOTEChandrasekharan1985108-5] 1 2 3 4 Chandrasekharan (1985), p. 108.

[FOOTNOTEChandrasekharan198563-6] Chandrasekharan (1985), p. 63.

[FOOTNOTEChandrasekharan1985117-7] Chandrasekharan (1985), p. 117.

[FOOTNOTERankin1977226–228-8] Rankin (1977), pp. 226–228.

[9] Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 103–109, 134

[Jacobi-11] 1 2 Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42

[13] Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 152

[14] Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. doi:10.1073/PNAS.35.7.371. PMC 1063041. S2CID 45071481.

[15] Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110.

[16] Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". Transactions of the American Mathematical Society. 349 (6): 2125–2173.

[17] Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240

[FOOTNOTEChandrasekharan1985121-18] Chandrasekharan (1985), p. 121.

[FOOTNOTEChandrasekharan1985118-19] Chandrasekharan (1985), p. 118.

[1] $λ (τ)$ is not a modular function (per the Wikipedia definition), but every modular function is a rational function in $λ (τ)$ . Some authors use a non-equivalent definition of "modular functions".

[10] For any prime power, we can iterate the modular equation of degree $p$ . This process can be used to give algebraic values of $λ (n i)$ for any $n \in ℕ .$

[12] $sl a ϖ$ is algebraic for every $a \in ℚ .$