Asymptotic and exact series representations for the incomplete Gamma function

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Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a,x) for any value of a or x. Applications of these formulas are discussed.