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 theory. 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 theory. For any complex number τ {displaystyle au } with I m ( τ ) > 0 {displaystyle mathrm {Im} ( au )>0} , let q = e 2 π i τ {displaystyle q=e^{2pi i au }} , then the eta function is defined by, The notation q ≡ e 2 π i τ {displaystyle qequiv e^{2pi { m {{i} au }}},} is now standard in number theory, though many older books use q for the nome e π i τ {displaystyle e^{pi { m {{i} au }}},} . 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. The eta function satisfies the functional equations More generally, suppose a, b, c, d are integers with ad − bc = 1, so that is a transformation belonging to the modular group. We may assume that either c > 0, or c = 0 and d = 1. Then