Cauchy’s formula for repeated integration is shown to be valid for the function
\[
R(t) = \lambda \Gamma(q) t^{q-1} E_{q,q}(-\lambda \Gamma(q)t^{q}),
\]
where $\lambda$ and $q$ are given positive constants with $q \in (0,1)$, $\Gamma$ is the Gamma function, and $E_{q,q}$ is a Mittag-Leffler function. The function $R$ is important in the study of Volterra integral equations because it is the unique continuous solution of the so-called resolvent equation
\[
R(t) = \lambda t^{q-1} – \lambda \int_{0}^{t} (t-s)^{q-1} R(s)\, ds
\]
on the interval $(0,\infty)$. This solution, commonly called the resolvent, is used to derive a formula for the unique continuous solution of the Riemann–Liouville fractional relaxation equation
\[
D^{q}x(t) = -a x(t) + g(t) \qquad (a > 0)
\]
on the interval $[0,\infty)$ when $g$ is a given polynomial. This formula is used to solve a generalization of the equation of motion of a falling body. The last example shows that the solution of a fractional relaxation equation may be quite elementary despite the complexity of the resolvent.