Contents Preface vii Chapter 1. The Gamma function extended to nonpositive integer points 1 1.1. Homogeneous distributions 1 1.2. A meromorphic extension of the Gamma function 4 1.3. Riesz regularisation 5 1.4. Hadamard’s “finite part” method 7 1.5. Discrepancies 12 Chapter 2. The canonical integral and noncommutative residue on symbols 15 2.1. Classical and log-polyhomogeneous symbols on Rd 15 2.2. The noncommutative residue on classical symbols 20 2.3. Closed linear forms on symbol valued forms 22 2.4. The noncommutative residue characterised via Stokes’ property. I 24 2.5. The canonical integral characterised via Stokes’ property. I 24 2.6. Characterisations by means of the translation invariance 28 2.7. The noncommutative residue characterised via its covariance. I 32 Chapter 3. The cut-off regularised integral 35 3.1. Cut-off (or Hadamard finite part) integral 35 3.2. Cut-off integrals and periods 38 3.3. Discrepancies of the cut-off integral 39 3.4. Characterisations by means of Stokes’ property. II 48 3.5. The canonical integral characterised by means of its covariance 49 Chapter 4. The noncommutative residue as a complex residue 53 4.1. Regularised evaluators 53 4.2. Meromorphic extensions of integrals on classical symbols 56 4.3. Extension to log-polyhomogeneous symbols 61 4.4. Invariance properties of the noncommutative residue 68 4.5. Dimensional versus cut-off regularised integrals 70 4.6. Discrepancies of regularised integrals 74 Chapter 5. The canonical sum on noninteger order classical symbols 79 5.1. The Euler-Maclaurin formula 79 5.2. The higher dimensional Euler-Maclaurin formula 83 5.3. Cut-off discrete sums on Zd subordinated to convex polytopes 86 5.4. Zd-translation invariant linear forms on symbols 91 5.5. The noncommutative residue and Zd-translation invariance 93 v
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