As defined by Erdélyi et al. (1981, p. 20), the G-function is given by G(z) congruent ψ_0(1/2 + 1/2 z) - ψ_0(1/2 z), where ψ_0(z) is the digamma function. Integral representations are given by G(z) | = | 2 integral_0^1 t^(z - 1)/(1 + t) d t | = | 2 integral_0^∞ e^(-z t)/(1 + e^(-t)) d t for ℜ[z]>0. G(z) is also given by the series G(z) = 2 sum_(n = 0)^∞ (-1)^n/(z + n), and in terms of the hypergeometric function by G(z) = 2z^(-1) _2 F_1(1, z;1 + z;-1).
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