Laguerre contour-integral amplitude
= Laguerre contour-integral amplitude
{c}
For the Laguerre equation
$$
zy''+(1-z)y'+\lambda y=0,
$$
the ansatz $y=\int_\gamma e^{zt}f(t)\,dt$ gives
$$
f(t)=t^{-\lambda-1}(t-1)^\lambda.
$$
The contour must make the endpoint contribution
$$
\left[e^{zt}t^{-\lambda}(t-1)^{\lambda+1}\right]_{\partial\gamma}
$$
vanish.