Laplace transform INDEX

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

Moreover, it is clear that such a relation also implies that the connected Green s functions are the Laplace transforms of connected partition functions (without index G). We call it the Laplace-de Gennes transformation. 

An integral transform is similar to a functional series, except that it contains an integration instead of a summation, which corresponds to an integration variable instead of a summation index. The integrand contains two factors, as does a term of a functional series. The first factor is the transform, which plays the same role as the coefficients of a power series. The second factor is the basis function, which plays the same role as the set of basis functions in a functional series. We discuss two types of transforms, Fourier transforms and Laplace transforms. 

In a similar fashion, one can write down the time-fractional Hamilton-Jacobi equation for the case where the waiting time PDF [Pg.162]

Laplace-de Gennes transform). This correspondence bet ween Green s functions and partition functions remains valid for their cumulants (also called connected parts). Thus, eqn (11.3.6) applies also to connected Green s functions and connected partition functions (the fact that they are connected is denoted by the absence of the index G). 

We are especially interested in the partition function +3T(S)= +3f(5,5 S) of an isolated polymer chain this function is calculated in the framework of the standard continuous model and the index + indicates that a short-range cut-off is introduced for regularization (see Chapter 10, Section 4.1). As was shown in Chapter 11, + (k, — k S) is the inverse Laplace-de Gennes transform of (k, — k a)... 

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