Actuarial values calculated using the incomplete Gamma function

The complete expectation-of-life for a person and the actuarial present value of continuous life annuities are defined by integrals. In all of them at least one of the factors is a survival function value ratio. If de Moivre’s law of mortality is chosen, such integrals can easily be evaluated; but if the Makeham survival function is adopted, they are used to be calculated numerically. For the above actuarial figures, closed form integrations are hereafter provided by means of the incomplete Gamma function.