The Dedekind eta function is defined over the upper half-plane H = {τ:ℑ[τ]>0} by η(τ) | congruent | (q^_)^(1/24) (q^_)_∞ | = | (q^_)^(1/24) product_(k = 1)^∞(1 - (q^_)^k) | = | (q^_)^(1/24) sum_(n = - ∞)^∞ (-1)^n (q^_)^(n(3n - 1)/2) | = | sum_(n = - ∞)^∞ (-1)^n (q^_)^((6n - 1)^2/24) | = | (q^_)^(1/24){1 + sum_(n = 1)^∞ (-1)^n[(q^_)^(n(3n - 1)/2) + (q^_)^(n(3n + 1)/2)]} | = | (q^_)^(1/24)(1 - q^_ - (q^_)^2 + (q^_)^5 + (q^_)^7 - (q^_)^12 - ...) (OEIS A010815), where q^_ congruent e^(2π i τ) is the square of the nome q, τ is the half-period ratio, and (q)_∞ is a q-series.
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