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Dedekind Eta Function

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The Dedekind eta function is defined over the upper half-plane H={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)

(OEIS A010815), where q^_=e^(2piitau) is the square of the nome q, tau is the half-period ratio, and (q)_infty is a q-series (Weber 1902, pp. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. 139).

The Dedekind eta function is implemented in the Wolfram Language as DedekindEta[tau].

Rewriting the definition in terms of q^_ explicitly in terms of the half-period ratio tau gives the product

eta(tau)=e^(piitau/12)product_(k=1)^infty(1-e^(2piiktau)).
(7)
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It is illustrated above in the complex plane.

eta(tau) is a modular form first introduced by Dedekind in 1877, and is related to the modular discriminant of the Weierstrass 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)

where zeta(z;g_2,g_3) is the Weierstrass zeta function and g_2 and g_3 are the invariants corresponding to the half-periods (1,tau). The derivative of eta(tau) satisfies

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

where G_2(tau) is an Eisenstein 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)

(OEIS A091343), where Gamma(z) is the gamma 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)

where P is the plastic constant, (P(x))_n denotes a polynomial root, and tau_0=(1+isqrt(23))/2.

Letting zeta_(24)=e^(2pii/24)=e^(pii/12) be a root 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)

where n is an integer (Weber 1902, p. 113; Atkin and Morain 1993; Apostol 1997, p. 47). The Dedekind eta function is related to the Jacobi theta function theta_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 expansions of the form (q,q)_infty^c to affine root systems. Exceptions not included in Macdonald's treatment include c=2, found by Hecke and Rogers, c=4, found by Ramanujan, and c=26, found by Atkin (Leininger and Milne 1999). Using the Dedekind eta function, the Jacobi 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, where Gamma is the modular group Gamma, c>0, and tau in H (where H is the upper 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 a Dedekind sum (Apostol 1997, pp. 52-57), with |_x_| the floor 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/

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References

Apostol, T. M. "The Dedekind Eta Function." Ch. 3 in Modular 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. Monthly 106, 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 in U(n)." Discr. Math. 204, 281-317, 1999a.Leininger, V. E. and Milne, S. C. "Some New Infinite Families of eta-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's eta-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 of eta(-1/tau)=eta(tau)sqrt(tau/i)." Mathematika 1, 4, 1954.Sloane, N. J. A. Sequences A010815, A091343, and A116397 in "The On-Line Encyclopedia of Integer Sequences."Weber, H. Lehrbuch der Algebra, Vols. I-III. 1902. Reprinted as Lehrbuch der Algebra, Vols. I-III, 3rd rev ed. New York: Chelsea, 1979.

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Dedekind Eta Function

Cite this as:

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

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