Laplace integral method for a differential equation

Laplace's integral method seeks $y(z)={\int }_C{e}^{zt}f(t)dt$. Substitution into the differential equation and integration by parts transfer multiplication by $z$ into differentiation with respect to $t$, producing a first-order equation for $f$ and endpoint conditions for $C$.

For $x>0$, substituting $y(x)={\int }_{\gamma }{e}^{xt}f(t)dt$ into $x{y}^{(3)}+2y=0$ and integrating by parts givesThe amplitude equation $(t^{3}f{)}^{\prime }=2f$ has solution $f(t)=C{t}^{-3}{e}^{-1/t^2}$. On $\gamma =(-\infty ,0)$ both endpoint terms vanish, and hence

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.

A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at $z=0$. It is therefore a constant multiple of the unique Frobenius power-series solution there.

For $\lambda =N\ge 0$, the amplitude $(t-1{)}^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda =-m<0$, the amplitude $t^{m-1}/(t-1{)}^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.

The normalized order-zero function has the representation