Expansion function damped exponential

At fixed 5, the terms in (5.180) represent the modes of the f-shape of the potential. The Bessel function oscillates and thus a s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance 5 the characteristic damping length is / = l/a , or, in the dimensional form... 

RhJiR) in terms of the complex zeros of l-p Cj(fe) = 0 (i.e., the poles kj. In general, an infinite number of poles contribute to this expansion, and some convergence problans related to the theoretical long tail of the direct correlation functions are encountered [185, 188, 200], However, the asymptotic behavior can be extracted by keeping only the pole =iy and the pair of lowest y -lying poles that are denoted conventionally by = x, +iy. Pure monotonic exponential decay or exponentially damped oscillatory decay can be identified from these poles. 

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