How should I regularize this integral?
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A regularization is just a way to extend the concept of convergence in a mathematical framework, nothing grants that the chosen.
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✍️Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by Hadamard (1923, book III, chapter I, 1932). Riesz (1938, 1949) showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.
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