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Dedekind Eta Function

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DedekindEta

The Dedekind eta function is defined over theupper half-planeH={tau:I[tau]>0} by

eta(tau)=q^_^(1/24)(q^_)_infty
(1)
=q^_^(1/24)product_(k=1)^(infty)(1-q^_^k)
(2)
=q^_^(1/24)sum_(n=-infty)^(infty)(-1)^nq^_^(n(3n-1)/2)
(3)
=sum_(n=-infty)^(infty)(-1)^nq^_^((6n-1)^2/24)
(4)
=q^_^(1/24){1+sum_(n=1)^(infty)(-1)^n[q^_^(n(3n-1)/2)+q^_^(n(3n+1)/2)]}
(5)
=q^_^(1/24)(1-q^_-q^_^2+q^_^5+q^_^7-q^_^(12)-...)
(6)

(OEISA010815), whereq^_=e^(2piitau) is the square of thenomeq,tau is thehalf-period ratio, and(q)_infty is aq-series (Weber 1902, pp. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. 139).

The Dedekind eta function is implemented in theWolframLanguage asDedekindEta[tau].

Rewriting the definition in terms ofq^_ explicitly in terms of thehalf-period ratiotau gives the product

eta(tau)=e^(piitau/12)product_(k=1)^infty(1-e^(2piiktau)).
(7)
DedekindEtaReIm
DedekindEtaContours

It is illustrated above in thecomplex plane.

eta(tau) is amodular form first introduced by Dedekind in 1877, and is related to themodular discriminant of theWeierstrass elliptic function by

Delta(tau)=(2pi)^(12)[eta(tau)]^(24)
(8)

(Apostol 1997, p. 47).

A compact closed form for the derivative is given by

(deta(tau))/(dtau)=i/pieta(tau)zeta(1;g_2,g_3),
(9)

wherezeta(z;g_2,g_3) is theWeierstrass zeta function andg_2 andg_3 are the invariants corresponding to the half-periods(1,tau). The derivative ofeta(tau) satisfies

-4piid/(dtau)ln[eta(tau)]=G_2(tau),
(10)

whereG_2(tau) is anEisenstein series, and

d/(dtau)ln[eta(-1/tau)]=d/(dtau)ln[eta(tau)]+1/2d/(dtau)ln(-itau).
(11)

A special value is given by

eta(i)=(Gamma(1/4))/(2pi^(3/4))
(12)
=0.7682254...
(13)

(OEISA091343), whereGamma(z) is thegamma function. Another special case is

P=(x^3-x-1)_1
(14)
=(e^(ipi/24)eta(tau_0))/(sqrt(2)eta(2tau_0))
(15)
=1.3247179572...
(16)

whereP is theplastic constant,(P(x))_n denotes apolynomial root, andtau_0=(1+isqrt(23))/2.

Lettingzeta_(24)=e^(2pii/24)=e^(pii/12) be aroot of unity,eta(tau) satisfies

eta(tau+1)=zeta_(24)eta(tau)
(17)
eta(tau+n)=zeta_(24)^neta(tau)
(18)
eta(-1/tau)=sqrt(-itau)eta(tau)
(19)

wheren is an integer (Weber 1902, p. 113; Atkin and Morain 1993; Apostol 1997, p. 47). The Dedekind eta function is related to theJacobi theta functiontheta_2 by

eta(q^_)=(theta_2(1/6pi,q^_^(1/6)))/(sqrt(3))
(20)

(Weber 1902, Vol. 3, p. 112) and

theta_3(0,e^(piitau))=(eta^2(1/2(tau+1)))/(eta(tau+1))
(21)

(Apostol 1997, p. 91).

Macdonald (1972) has related most expansionsof the form(q,q)_infty^c to affineroot systems. Exceptions not included in Macdonald's treatment includec=2, found by Hecke and Rogers,c=4, found by Ramanujan, andc=26, found by Atkin (Leininger and Milne 1999). Using the Dedekind eta function, theJacobi triple product identity

(q,q)_infty^3=sum_(n=0)^infty(-1)^n(2n+1)q^(n(n+1)/2)
(22)

can be written

eta^3(tau)=sum_(n=0)^infty(-1)^n(2n+1)q^_^((2n+1)^2/8)
(23)

(Jacobi 1829, Hardy and Wright 1979, Hirschhorn 1999, Leininger and Milne 1999).

Dedekind's functional equation states that if[a b; c d] in Gamma, whereGamma is themodular group Gamma,c>0, andtau in H (whereH is theupper half-plane), then

eta((atau+b)/(ctau+d))=epsilon(a,b,c,d)[sqrt(-i(ctau+d))]eta(tau),
(24)

where

epsilon(a,b,c,d)=exp[pii((a+d)/(12c)+s(-d,c))],
(25)

and

s(h,k)=sum_(r=1)^(k-1)r/k((hr)/k-|_(hr)/k_|-1/2)
(26)

is aDedekind sum (Apostol 1997, pp. 52-57), with|_x_| thefloor function.

See also

Dirichlet Eta Function,Dedekind Sum,Elliptic Invariants,Elliptic Lambda Function,Infinite Product,Jacobi Theta Functions,Klein's Absolute Invariant,q-Product,q-Series,Rogers-Ramanujan Continued Fraction,Tau Function,Weber Functions

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/DedekindEta/

Explore with Wolfram|Alpha

References

Apostol, T. M. "The Dedekind Eta Function." Ch. 3 inModular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 47-73, 1997.Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primality Proving."Math. Comput.61, 29-68, 1993.Berndt, B. C.Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Bhargava, S. and Somashekara, D. "Some Eta-Function Identities Deducible from Ramanujan's_1psi_1 Summation."J. Math. Anal. Appl.176, 554-560, 1993.Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Hirschhorn, M. D. "Another Short Proof of Ramanujan's Mod 5 Partition Congruences, and More."Amer. Math. Monthly106, 580-583, 1999.Jacobi, C. G. J.Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, p. 90, 1829.Leininger, V. E. and Milne, S. C. "Expansions for(q)_infty^(n^2+n) and Basic Hypergeometric Series inU(n)."Discr. Math.204, 281-317, 1999a.Leininger, V. E. and Milne, S. C. "Some New Infinite Families ofeta-Function Identities."Methods Appl. Anal.6, 225-248, 1999b.Köhler, G. "Some Eta-Identities Arising from Theta Series."Math. Scand.66, 147-154, 1990.Macdonald, I. G. "Affine Root Systems and Dedekind'seta-Function."Invent. Math.15, 91-143, 1972.Ramanujan, S. "On Certain Arithmetical Functions."Trans. Cambridge Philos. Soc.22, 159-184, 1916.Siegel, C. L. "A Simple Proof ofeta(-1/tau)=eta(tau)sqrt(tau/i)."Mathematika1, 4, 1954.Sloane, N. J. A. SequencesA010815,A091343, andA116397 in "The On-Line Encyclopedia of Integer Sequences."Weber, H.Lehrbuch der Algebra, Vols. I-III. 1902. Reprinted asLehrbuch der Algebra, Vols. I-III, 3rd rev ed. New York: Chelsea, 1979.

Referenced on Wolfram|Alpha

Dedekind Eta Function

Cite this as:

Weisstein, Eric W. "Dedekind Eta Function."FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/DedekindEtaFunction.html

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Created, developed and nurtured by Eric Weisstein at Wolfram Research
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