I suppose it is a condition for convergence of the integral. A simple example of divergence:
With[{t = 0, u = 0, a = 0},
Integrate[Exp[I x t - a (u - x)], {x, -Infinity, u}]]
We may find more examples this way:
sol = FindInstance[Not[Re[a] > Im[t] &&
Re[a (1000076 + u)] >= 1000076 Im[t] &&
Re[a (1000303 + u)] >= 1000303 Im[t]],
{t, u, a}, 2]
Exp[I x t - a (u - x)] /. sol[[1]] // ComplexExpand
