Dedekind Eta Function
The Dedekind eta function is defined over the upper half-plane ![H=<span style=]()
{tau:I[tau]>0}" src="http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/Inline1.gif" style="height:14px; width:101px" /> by
(OEIS A010815), where 
is the square of the nome 
, 
is the half-period ratio, and 
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 
explicitly in terms of the half-period ratio 
gives the product
 |
(7)
|
It is illustrated above in the complex plane.
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)](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/NumberedEquation2.gif) |
(8)
|
(Apostol 1997, p. 47).
A compact closed form for the derivative is given by
 |
(9)
|
where 
is the Weierstrass zeta function and 
and 
are the invariants corresponding to the half-periods 
. The derivative of 
satisfies
![-4piid/(dtau)ln[eta(tau)]=G_2(tau),](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/NumberedEquation4.gif) |
(10)
|
where 
is an Eisenstein series, and
![d/(dtau)ln[eta(-1/tau)]=d/(dtau)ln[eta(tau)]+1/2d/(dtau)ln(-itau).](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/NumberedEquation5.gif) |
(11)
|
A special value is given by
(OEIS A091343), where 
is the gamma function. Another special case is
where 
is the plastic constant, 
denotes a polynomial root, and 
.
Letting 
be a root of unity, 
satisfies
where 
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 
by
 |
(20)
|
(Weber 1902, Vol. 3, p. 112) and
 |
(21)
|
(Apostol 1997, p. 91).
Macdonald (1972) has related most expansions of the form 
to affine root systems. Exceptions not included in Macdonald's treatment include 
, found by Hecke and Rogers, 
, found by Ramanujan, and 
, found by Atkin (Leininger and Milne 1999). Using the Dedekind eta function, the Jacobi triple product identity
 |
(22)
|
can be written
 |
(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](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/Inline69.gif)
, where 
is the modular group Gamma, 
, and 
(where 
is the upper half-plane), then
![eta((atau+b)/(ctau+d))=epsilon(a,b,c,d)[sqrt(-i(ctau+d))]eta(tau),](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/NumberedEquation10.gif) |
(24)
|
where
![epsilon(a,b,c,d)=exp[pii((a+d)/(12c)+s(-d,c))],](http://mathworld.wolfram.com/images/equations/DedekindEtaFunction/NumberedEquation11.gif) |
(25)
|
and
 |
(26)
|
is a Dedekind sum (Apostol 1997, pp. 52-57), with 
the floor function.
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 
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 
and Basic Hypergeometric Series in 
." Discr. Math. 204, 281-317, 1999a.
Leininger, V. E. and Milne, S. C. "Some New Infinite Families of 
-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 
-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 
." 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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