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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the modular lambda function λ(τ)modular function In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
(per the Wikipedia definition), but every modular function is a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in \lambda(\tau). Some authors use a non-equivalent definition of "modular functions". is a highly symmetric holomorphic function on the complex
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the
modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular grou ...
''X''(2). Over any point τ, its value can be described as a
cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
of the branch points of a ramified double cover of the projective line by the
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
\mathbb/\langle 1, \tau \rangle, where the map is defined as the quotient by the minus;1involution. The q-expansion, where q = e^ is the nome, is given by: : \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots. By symmetrizing the lambda function under the canonical action of the symmetric group ''S''3 on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group \operatorname_2(\mathbb), and it is in fact Klein's modular
j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such funct ...
.


Modular properties

The function \lambda(\tau) is invariant under the group generated byChandrasekharan (1985) p.115 : \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac \ . The generators of the modular group act byChandrasekharan (1985) p.109 : \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac \, ; : \tau \mapsto -\frac \ :\ \lambda \mapsto 1 - \lambda \ . Consequently, the action of the modular group on \lambda(\tau) is that of the
anharmonic group In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
, giving the six values of the
cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
:Chandrasekharan (1985) p.110 : \left\lbrace \right\rbrace \ .


Relations to other functions

It is the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
of the elliptic modulus,Chandrasekharan (1985) p.108 that is, \lambda(\tau)=k^2(\tau). In terms of the
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
\eta(\tau) and
theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
s, : \lambda(\tau) = \Bigg(\frac\Bigg)^8 = \frac =\frac and, : \frac-\big(\lambda(\tau)\big)^ = \frac\left(\frac\right)^4 = 2\,\frac whereChandrasekharan (1985) p.63 :\theta_2(\tau) =\sum_^\infty e^ :\theta_3(\tau) = \sum_^\infty e^ :\theta_4(\tau) = \sum_^\infty (-1)^n e^ In terms of the half-periods of
Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by th ...
, let omega_1,\omega_2/math> be a
fundamental pair of periods In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definition ...
with \tau=\frac. : e_1 = \wp\left(\frac\right), \quad e_2 = \wp\left(\frac\right),\quad e_3 = \wp\left(\frac\right) we have : \lambda = \frac \, . Since the three half-period values are distinct, this shows that \lambda does not take the value 0 or 1. The relation to the
j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such funct ...
isChandrasekharan (1985) p.117 : j(\tau) = \frac = \frac \ . which is the ''j''-invariant of the elliptic curve of
Legendre form In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Adrien-Marie Legendre, Legendre chose the name ''elliptic integrals'' because the second kind gives the ar ...
y^2=x(x-1)(x-\lambda)


Modular equations

The ''modular equation of degree'' p (where p is a prime number) is an algebraic equation in \lambda (p\tau) and \lambda (\tau). If \lambda (p\tau)=u^8 and \lambda (\tau)=v^8, the modular equations of degrees p=2,3,5,7 are, respectively, :(1+u^4)^2v^8-4u^4=0, :u^4-v^4+2uv(1-u^2v^2)=0, :u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0, :(1-u^8)(1-v^8)-(1-uv)^8=0. The quantity v (and hence u) can be thought of as a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
on the upper half-plane \operatorname\tau>0: :\beginv&=\prod_^\infty \tanh\frac=\sqrte^\frac\\ &=\cfrac\end Since \lambda(i)=1/2, the modular equations can be used to give algebraic values of \lambda(pi) for any prime p.For any
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
, we can iterate the modular equation of degree p. This process can be used to give algebraic values of \lambda (ni) for any n\in\mathbb. The algebraic values of \lambda(ni) are also given by p. 42\operatornamea\varpi is algebraic for every a\in\mathbb. :\lambda (ni)=\prod_^ \operatorname^8\frac\quad (n\,\text) :\lambda (ni)=\frac\prod_^ \left(1-\operatorname^2\frac\right)^2\quad (n\,\text) where \operatorname is the lemniscate sine and \varpi is the
lemniscate constant In mathematics, the lemniscate constant p. 199 is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of for the circle. Equivalently, the perimeter ...
.


Lambda-star


Definition and computation of lambda-star

The function \lambda^*(x) (where x\in\mathbb^+) 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) are related by following expression: :\frac = \sqrt The values of \lambda^*(x) can be computed as follows: :\lambda^*(x) = \frac :\lambda^*(x) = \left sum_^\infty\exp[-(a+1/2)^2\pi\sqrtright.html" ;"title="(a+1/2)^2\pi\sqrt.html" ;"title="sum_^\infty\exp[-(a+1/2)^2\pi\sqrt">sum_^\infty\exp[-(a+1/2)^2\pi\sqrtright">(a+1/2)^2\pi\sqrt.html" ;"title="sum_^\infty\exp[-(a+1/2)^2\pi\sqrt">sum_^\infty\exp[-(a+1/2)^2\pi\sqrtright2\left[\sum_^\infty\exp(-a^2\pi\sqrt)\right]^ :\lambda^*(x) = \left[\sum_^\infty\operatorname[(a+1/2)\pi\sqrt]\right]\left[\sum_^\infty\operatorname(a\pi\sqrt)\right]^ The functions \lambda^* and \lambda are related to each other in this way: :\lambda^*(x) = \sqrt


Properties of lambda-star

Every \lambda^* value of a positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
is a positive
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
: :\lambda^*(x \in \mathbb^+) \in \mathbb^+. K(\lambda^*(x)) and E(\lambda^*(x)) (the
complete elliptic integral of the second kind In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
) can be expressed in closed form in terms of the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
for any x\in\mathbb^+, as Selberg and Chowla proved in 1949. The following expression is valid for all n \in \mathbb: :\sqrt = \sum_^ \operatorname\left fracK\left[\lambda^*\left(\frac\right)\right\lambda^*\left(\frac\right)\right.html" ;"title="lambda^*\left(\frac\right)\right.html" ;"title="fracK\left[\lambda^*\left(\frac\right)\right">fracK\left[\lambda^*\left(\frac\right)\right\lambda^*\left(\frac\right)\right">lambda^*\left(\frac\right)\right.html" ;"title="fracK\left[\lambda^*\left(\frac\right)\right">fracK\left[\lambda^*\left(\frac\right)\right\lambda^*\left(\frac\right)\right where \operatorname is the Jacobi elliptic function delta amplitudinis with modulus k. By knowing one \lambda^* value, this formula can be used to compute related \lambda^* values: :\lambda^*(n^2x) = \lambda^*(x)^n\prod_^\operatorname\left\^2 where n\in\mathbb and \operatorname is the Jacobi elliptic function sinus amplitudinis with modulus k. Further relations: :\lambda^*(x)^2 + \lambda^*(1/x)^2 = 1 : lambda^*(x)+1\lambda^*(4/x)+1] = 2 :\lambda^*(4x) = \frac = \tan\left\^2 :\lambda^*(x) - \lambda^*(9x) = 2 lambda^*(x)\lambda^*(9x) - 2 lambda^*(x)\lambda^*(9x) \begin & a^-f^ = 2af +2a^5f^5\, &\left(a = \left frac\right\right) &\left(f = \left frac\right\right) \\ &a^+b^-7a^4b^4 = 2\sqrtab+2\sqrta^7b^7\, &\left(a = \left frac\right\right) &\left(b = \left frac\right\right) \\ & a^-c^ = 2\sqrt(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left frac\right\right) &\left(c = \left frac\right\right) \\ & (a^2-d^2)(a^4+d^4-7a^2d^2) a^2-d^2)^4-a^2d^2(a^2+d^2)^2= 8ad+8a^d^\, &\left(a = \left frac\right\right) &\left(d = \left frac\right\right) \end Lambda-star values of integer numbers of 4n-3-type: :\lambda^*(1) = \frac :\lambda^*(5) = \sin\left frac\arcsin\left(\sqrt-2\right)\right/math> :\lambda^*(9) = \frac(\sqrt-1)(\sqrt-\sqrt :\lambda^*(13) = \sin\left frac\arcsin(5\sqrt-18)\right/math> :\lambda^*(17) = \sin\left\ :\lambda^*(21) = \sin\left\ :\lambda^*(25) = \frac(\sqrt-2)(3-2\sqrt :\lambda^*(33) = \sin\left\ :\lambda^*(37) = \sin\left\ :\lambda^*(45) = \sin\left\ :\lambda^*(49) = \frac(8+3\sqrt)(5-\sqrt-\sqrt \left(\sqrt-\sqrt-\sqrt sqrt\right) :\lambda^*(57) = \sin\left\ :\lambda^*(73) = \sin\left\ Lambda-star values of integer numbers of 4n-2-type: :\lambda^*(2) = \sqrt-1 :\lambda^*(6) = (2-\sqrt)(\sqrt-\sqrt) :\lambda^*(10) = (\sqrt-3)(\sqrt-1)^2 :\lambda^*(14) = \tan\left\ :\lambda^*(18) = (\sqrt-1)^3(2-\sqrt)^2 :\lambda^*(22) = (10-3\sqrt)(3\sqrt-7\sqrt) :\lambda^*(30) = \tan\left\ :\lambda^*(34) = \tan\left\ :\lambda^*(42) = \tan\left\ :\lambda^*(46) = \tan\left\ :\lambda^*(58) = (13\sqrt-99)(\sqrt-1)^6 :\lambda^*(70) = \tan\left\ :\lambda^*(78) = \tan\left\ :\lambda^*(82) = \tan\left\ Lambda-star values of integer numbers of 4n-1-type: :\lambda^*(3) = \frac(\sqrt-1) :\lambda^*(7) = \frac(3-\sqrt) :\lambda^*(11) = \frac(\sqrt+3)\left(\frac\sqrt \frac\sqrt \frac\sqrt-1\right)^4 :\lambda^*(15) = \frac(3-\sqrt)(\sqrt-\sqrt)(2-\sqrt) :\lambda^*(19) = \frac(3\sqrt+13)\left \frac(\sqrt-2-\sqrt)\sqrt_\frac(5-\sqrt)\right.html" ;"title="frac(\sqrt-2+\sqrt)\sqrt \frac(\sqrt-2-\sqrt)\sqrt \frac(5-\sqrt)\right">frac(\sqrt-2+\sqrt)\sqrt \frac(\sqrt-2-\sqrt)\sqrt \frac(5-\sqrt)\right4 :\lambda^*(23) = \frac(5+\sqrt)\left \frac(\sqrt-1)\sqrt_\frac\right.html" ;"title="frac(\sqrt+1)\sqrt \frac(\sqrt-1)\sqrt \frac\right">frac(\sqrt+1)\sqrt \frac(\sqrt-1)\sqrt \frac\right4 :\lambda^*(27) = \frac(\sqrt-1)^3\left[\frac\sqrt(\sqrt \sqrt 1)-\sqrt 1\right]^4 :\lambda^*(39) = \sin\left\ :\lambda^*(55) = \sin\left\ Lambda-star values of integer numbers of 4n-type: :\lambda^*(4) = (\sqrt-1)^2 :\lambda^*(8) = \left(\sqrt+1-\sqrt\right)^2 :\lambda^*(12) = (\sqrt-\sqrt)^2(\sqrt-1)^2 :\lambda^*(16) = (\sqrt+1)^2(\sqrt 1)^4 :\lambda^*(20) = \tan\left frac\arcsin(\sqrt-2)\right2 :\lambda^*(24) = \tan\left\^2 :\lambda^*(28) = (2\sqrt-\sqrt)^2(\sqrt-1)^4 :\lambda^*(32) = \tan\left\^2 Lambda-star values of rational fractions: :\lambda^*\left(\frac\right) = \sqrt :\lambda^*\left(\frac\right) = \frac(\sqrt+1) :\lambda^*\left(\frac\right) = (2-\sqrt)(\sqrt+\sqrt) :\lambda^*\left(\frac\right) = 2\sqrt \sqrt-1) :\lambda^*\left(\frac\right) = \sqrt \sqrt-\sqrt)(\sqrt+1)\sqrt :\lambda^*\left(\frac\right) = \frac\left(\sqrt+\sqrt-1\right) :\lambda^*\left(\frac\right) = (\sqrt-3)(\sqrt+1)^2 :\lambda^*\left(\frac\right) = \frac(3+\sqrt)(\sqrt-\sqrt)(2+\sqrt) :\lambda^*\left(\frac\right) = \tan\left frac-\frac\arcsin(\sqrt-2)\right2


Ramanujan's class invariants

Ramanujan's class invariants G_n and g_n are defined as :G_n=2^e^\prod_^\infty \left(1+e^\right), :g_n=2^e^\prod_^\infty \left(1-e^\right), where n\in\mathbb^+. For such n, the class invariants are algebraic numbers. For example :g_=\sqrt, \quad g_=\sqrt. Identities with the class invariants include :G_n=G_,\quad g_=\frac,\quad g_=2^g_nG_n. The class invariants are very closely related to the Weber modular functions \mathfrak and \mathfrak_1. These are the relations between lambda-star and the class invariants: :G_n = \sin\^ = 1\Big /\left sqrt[12sqrt[24.html"_;"title="2.html"_;"title="sqrt[12">sqrt[12sqrt[24">2.html"_;"title="sqrt[12">sqrt[12sqrt[24right.html" ;"title="2">sqrt[12sqrt[24.html" ;"title="2.html" ;"title="sqrt[12">sqrt[12sqrt[24">2.html" ;"title="sqrt[12">sqrt[12sqrt[24right">2">sqrt[12sqrt[24.html" ;"title="2.html" ;"title="sqrt[12">sqrt[12sqrt[24">2.html" ;"title="sqrt[12">sqrt[12sqrt[24right :g_n = \tan\^ = \sqrt[12] :\lambda^*(n) = \tan\left\ = \sqrt-g_n^


Other appearances


Little Picard theorem

The lambda function is used in the original proof of the
Little Picard theorem In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
, that an
entire Entire may refer to: * Entire function, a function that is holomorphic on the whole complex plane * Entire (animal), an indication that an animal is not neutered * Entire (botany) This glossary of botanical terms is a list of definitions of ...
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,∞. Consider the function ''z'' → ω(''f''(''z'')). By the
Monodromy theorem In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply ''analytic fun ...
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.Chandrasekharan (1985) p.118


Moonshine

The function \tau\mapsto 16/\lambda(2\tau) - 8 is the normalized
Hauptmodul In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a Quotient by a group action, quotient of the complex upper half-plane H by the Group action (mathematics) ...
for the group \Gamma_0(4), and its ''q''-expansion q^ + 20q - 62q^3 + \dots, where q=e^, is the graded character of any element in conjugacy class 4C of the
monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order    246320597611213317192329314147 ...
acting on the
monster vertex algebra The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by ap ...
.


Footnotes


References


Notes


Other

* * * * * * 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


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