= Retarded fundamental solution of a constant-coefficient ordinary differential operator
{title2=$E=Hu,\quad P(D)E=\delta_0$}
For $P(z)=a_Nz^N+\cdots+a_0$ with $N\geq1$ and $a_N\ne0$, the unique <retarded fundamental solution> is $E=Hu$, where $P(D)u=0$, $u^{(j)}(0)=0$ for $j<N-1$, and $u^{(N-1)}(0)=1/a_N$. The <distributional jump formula for a Heaviside product> gives $P(D)E=\delta_0$. Its explicit expression is
$$
u(x)=\sum_{P(\lambda)=0}\operatorname*{Res}_{z=\lambda}\frac{e^{zx}}{P(z)}.
$$
The <residue theorem> gives the initial derivatives from the coefficients at infinity. Equivalently, the inverse <Fourier transform> is integrated below all its poles, producing <support of a distribution> in $[0,\infty)$. Uniqueness follows because the difference of two retarded solutions is an analytic homogeneous solution vanishing on the negative half-line.
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