= Adler trace
{c}
{title2=$\operatorname{Tr}P=\int\operatorname{res}P\,dx$}
Integrate the <formal pseudodifferential residue> over a period, or over the real line when the integral converges and the relevant boundary terms vanish. More algebraically, apply a linear <functional> on the <coefficient> algebra that annihilates <derivatives>. For $A=a\partial^p$, $B=b\partial^q$ and $r=p+q+1\geq0$, the two residues are $\binom pr ab^{(r)}$ and $\binom qr ba^{(r)}$. Since $\binom qr=(-1)^r\binom pr$, <integration by parts> proves $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$. For $r<0$ both residues vanish. Bounded orders ensure only finitely many monomial pairs contribute to the residue of a product.
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