A Laurent expansion for regularised integrals of holomorphic symbols

TL;DR

This paper derives explicit formulas for all coefficients in the Laurent expansion of the Kontsevich-Vishik trace for holomorphic families of pseudodifferential operators, extending known residue results to higher orders.

Contribution

It provides a comprehensive Laurent expansion formula for the canonical trace, generalizing the Wodzicki residue to all higher order terms.

Findings

Explicit Laurent expansion coefficients for the canonical trace

Generalization of Wodzicki residue to higher order terms

Exact formulas for all coefficients in the expansion

Abstract

For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.