Abstract
In this chapter we introduce a number of mathematical tools, which will prove useful in the spectral action computations. Firstly, we consider the basic properties of some spectral functions via the functional calculus and general Dirichlet series. Next, we study the interplay between these provided by the functional transforms of Mellin and Laplace. The remainder of the chapter is devoted to various notions from the theory of asymptotic behaviour of functions and distributions.
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Notes
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- 2.
Strictly speaking, Formula (1.46) requires f to be the Laplace transform of a signed measure, which is not the case for the counting function. Nevertheless, naively \(\int _0^1 u^{k-1}\,du = 1/k\).
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Eckstein, M., Iochum, B. (2018). The Toolkit for Computations. In: Spectral Action in Noncommutative Geometry. SpringerBriefs in Mathematical Physics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-94788-4_2
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