= Solution
Write $\lambda=-m$ with $m\in\mathbb N$. The amplitude is now the rational function
$$
f(t)=\frac{t^{m-1}}{(t-1)^m}.
$$
Choose $\gamma_1$ 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 is
$$
y_{\gamma_1}(z)
=\frac{2\pi i}{(m-1)!}
\left.\frac{d^{m-1}}{dt^{m-1}}
\left(e^{zt}t^{m-1}\right)\right|_{t=1}.
$$
Every derivative term contains $e^z$ times a power of $z$ of degree at most $m-1$, and the highest-degree term is nonzero. Hence
$$
\boxed{y_1(z)=e^zP_{m-1}(z)}
$$
up 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>.
Back to article page
