The motivation of this work is to search for an elementary common physical mechanism for the non-exponential decay observed in a large number of disparate phenomena. Probably the most elementary assumption is an initial Poissonian distribution of activation energies E with each fraction decaying independently according to an Arrhenius rate r(E) approximately exp(-E/T). Despite this drastic oversimplification, this elementary model will account for a large variety of observed shapes of decay, ranging from nearly exponential to power law and nearly logarithmic functions. The fundamental feature that governs the shape is the ratio E/T, denoted 1/b, of the mean E at temperature T. Analysing limited experimental data sets is also easy due to the closed form of the resulting normalized decay curves g(b, tau )=b tau -b gamma (b,tau/), where tau =r0t is the time t normalized by the rate r0. The incomplete gamma function gamma (b, tau )= integral 0tau sb-1e-sds is an important correction of the power law tau -b In an alternative interpretation, this decay form g(b, tau ) could also be considered as generated by a b-dependent form of an effective barrier energy Ueff(g) increasing as a function of the decaying normalized observable g. In this paper the forms g(b, tau ) and Ueff(g) are evaluated, displayed and compared with In a separate paper the widespread applicability of the form g(b,tau/) will be demonstrated by interpreting various types of measured data using individual established temperature dependencies and by treating the 'ageing' or 'memory' effect.