 Solution

Write $\lambda =-m$ with $m\in \mathbb{N}$. The amplitude is now the rational functionChoose ${\gamma }_1$ to be a small positively oriented circle around the pole $t=1$, and choose ${\gamma }_2$ from $t=0$ along the negative real axis to $-\infty$. At zero the endpoint factor behaves as $t^m$ and vanishes; at negative infinity the exponential decays.

By the residue theorem, the finite-contour solution isEvery derivative term contains $e^z$ times a power of $z$ of degree at most $m-1$, and the highest-degree term is nonzero. Henceup to a constant, where $P_{m-1}$ is a polynomial of degree $m-1$. This is the negative-integer case of the Integer-parameter Laguerre contour residues.