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In this article, we obtain simplified as well as original closed-form expressions for integrals involving the gamma function. These explicit results are found neither in the literature nor using symbolic computation software. Subsequent results follow, giving rise to explicit expressions for integrals involving the error function, with application in order statistics. In particular, these results can be used within the framework of reliability theory.

Error backpropagation in networks of spiking neurons (SpikeProp) shows promise for the supervised learning of temporal patterns. However, its widespread use is hindered by its computational load and occasional convergence failures. In this letter, we show that the neuronal firing time equation at the core of SpikeProp can be solved analytically using the Lambert W function, offering a marked reduction in execution time over the step-based method used in the literature. Applying this analytical method to SpikeProp, we find that training time per epoch can be reduced by 12% to 56% under different experimental conditions. Finally, this work opens the way for further investigations of SpikeProp’s convergence behavior.