= Gaussian drift primitives
{c}
{title2=$E_0(z),\ E_1(z)$}
Define $E_0(z)=\int_0^z e^{-t^2}\int_0^t e^{u^2}\,du\,dt$ and $E_1(z)=\int_0^z e^{-t^2}\int_0^t e^{u^2}\operatorname{erf}u\,du\,dt$. They satisfy $(D^2+2zD)E_0=1$ and $(D^2+2zD)E_1=\operatorname{erf}z$. The first is an <even function>, the second an <odd function>, and their real tails are respectively $\frac12\log|z|+C_1+o(1)$ and $\operatorname{sgn}z[\frac12\log|z|+C_2+o(1)]$. Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a <singular perturbation>.
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