Numerical analytic continuation of Euclidean data
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To entirely determine the resulting functions of one-loop integrals it is necessary to find the correct analytic continuation to all relevant kinematical regions. We argue that this continuation procedure may be performed in a general and mathematical accurate way by using the ${\cal R}$ function notation of these integrals. The two- and three-point cases are discussed explicitly in this manner.
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We prove the analytic continuation of a certain family of Whittaker Archimedean integrals that arise as local factors of global $L$-functions associated to the standard representation of certain classical groups.
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In this paper a method is devised for the continuation into the discrete plane Q' of functions defined on the positive half-axes and the properties of continuation operator discussed.
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We will now show that the "discrete" and "continuous" variants of analytic continuation given in Definitions 19.3 and 19.4, respectively, are closely related. The fact stated below is not required for our proof of Theorem 8.3 later in the course, but we have chosen to include it in the lecture in order to further clarify the concept of analytic continuation.
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The problem of analytic continuation to the boundary of the holomorphy domain from both continuous and discrete interior sets has recently been the subject of detailed analyses. This problem is important in phenomenological applications but is also of interest in theoretical calculations, e.g., in attempting to evaluate the parameters of resonances or other nonperturbative effects in QCD. Because of the inherent instability of the continuation problem it is necessary to introduce additional criteria—which should be physically based—to select the right continuation function. In this paper, the results thus obtained for continuation from a continuum are examined for stability, and bounds are derived for the errors on the boundary in terms of the uncertainty of the input data. The procedure is shown to be stable in the sense that these bounds tend to zero as the data errors go to zero.
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The second group of results that we need to obtain Theorem 8.3 concerns analytic continuation. In fact, as explained in Remark 20.1, we could avoid using analytic continuation for the purposes of proving Theorem 8.3. However, this is an important concept that is widely used in complex analysis in general, and we feel that it should be part of our presentation.
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