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Efficient dispersion relations for terahertz spectroscopy
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14.It can be shown (Ref. 2) that the response function (e.g., the linear susceptibility or complex reflection coefficient) satisfies the dispersion relations if its analytic continuation into the upper half of the complex frequency plane (i) satisfies the condition at , (ii) has no poles in upper half plane, and (iii) fulfills the symmetry relation .
15.One can observe that a function, which can be presented as a sum of response functions, is also a response function. Hence in order to establish that the introduced polynomial is a response function, i.e., SSKK relations are valid for it, we have to show that functions , where . and is a constant, are response functions. It has been already shown that is a response function when (Refs. 10 and 11). Thus from the fact that multiplication of function by a real number will not affect requirements [(i), (ii), and (iii)] we can conclude then that is a response function and it turns out that SSKK relations are valid for the polynomial of Eq. (2).
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