= Airy resolvent kernel
{c}
{title2=$R_p(s)$}
For $\operatorname{Re}p>0$, put $\lambda=p^{1/3}$ using the <principal cube root> and $r_{1,2}=\lambda e^{\pm i\pi/3}$. The decaying <Green function> for $\partial_s^3+p$ on the real line is
$$
R_p(s)=\begin{cases}e^{-\lambda s}/(3\lambda^2),&s\geq0,\\-\sum_{j=1}^2e^{r_js}/(3r_j^2),&s<0.\end{cases}
$$
The identities among the three <characteristic roots> show that $R_p$ and $R_p'$ are continuous, while $R_p''$ has jump one. Its <distributional derivative> therefore satisfies $(\partial_s^3+p)R_p=\delta_0$. This is the kernel of the <resolvent operator> $(p+\partial_s^3)^{-1}$, obtained by the time <Laplace transform> of the <Airy equation>.
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