For any complex number τ with Im(τ) > 0, let q = e2πiτ; then the eta function is defined by,
The notation q ≡ e2πiτ is now standard in number theory, though many older books use q for the nomeeπiτ. Raising the eta equation to the 24th power and multiplying by (2π)12 gives
where Δ is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice.
The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.
Modulus of Euler phi on the unit disc, colored so that black = 0, red = 4
The real part of the modular discriminant as a function of q.
Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms. In particular the modular discriminant of Weierstrass can be defined as
and is a modular form of weight 12. Some authors omit the factor of (2π)12, so that the series expansion has integral coefficients.
Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
The picture on this page shows the modulus of the Euler function: the additional factor of q1/24 between this and eta makes almost no visual difference whatsoever (it only introduces a tiny pinprick at the origin). Thus, this picture can be taken as a picture of eta as a function of q.
Combinatorial identities[]
The theory of the algebraic characters of the affine Lie algebras gives rise to a large class of previously unknown identities for the eta function. These identities follow from the Weyl–Kac character formula, and more specifically from the so-called "denominator identities". The characters themselves allow the construction of generalizations of the Jacobi theta function which transform under the modular group; this is what leads to the identities. An example of one such new identity[3] is
where q = e2πiτ is the q-analog or "deformation" of the highest weight of a module.
Special values[]
From the above connection with the Euler function together with the special values of the latter, it can be easily deduced that
Eta quotients[]
Eta quotients are defined by quotients of the form
where d is a non-negative integer and rd is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may be algebraic, while combinations of eta quotients may even be integral. For example, define,