Consider the recursive definition of the following function:
\begin{equation} \omega(t, a) = 1 + \frac{1}{t} \cdot \biggl( \int_{0}^{t-a}\! \omega(t-x, a) \, dx \biggr) \end{equation}
We can assume that $t \geq a$.
The base branch is not given, but we can infer from the recursive branch. When $t = a$, $\omega (t, a) = \omega(a, a) = 1$.
It would be best to write the analytic expression for $\omega(t, a)$. However, if there is no way to accurately write the analytic expression of $\omega(t, a)$, Is there any approximation method to approaching its value in any precision?