Fourier-Laplace transform

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

As neutron scattering does not distinguish between the molecular levels, one wants to know the probability density in space Gs(r, t) = P r, t) + P2(r, t). Its Fourier-Laplace transform is... 

The rest of the analysis is now carried out, using the formalism presented by Leutheusser [34]. Let us introduce a Fourier-Laplace transform or onesided Fourier transform,... 

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

self-adjoint problems D(W) = D(H) and for bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

To connect the equations in (B.l) through the Fourier-Laplace transform, we need to define suitable complex contours to make the transforms convergent. Specifically we identify the contours C by the lines in upper and lower complex planes defined by CU ( id — oo — id + oo), where d > 0 may be arbitrary. Using the Heaviside function, 0(f), and the Dirac delta function, 5(f), we can characterize positive and negative times (with respect to f = 0) as linked with appropriate contours C as... 

The exact time evolution within subspace O follows from the convolution theorem of the Fourier-Laplace transform, i.e.,... 

By using time-to-frequency Fourier-Laplace transforms ... 

In the case of a polar liquid at low k(k = k ), these components can be related to Fourier-Laplace transforms of the time correlations of the appropriate components of the collective dipole density... 

The components of the permittivity tensor are related to Fourier-Laplace transforms, d>MA(k, to), of these TCFs by... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

The physical meaning of the assumption of Eq. (112) is evident. We are setting the condition that in this case persistence refers to velocity rather than position, while the division by (f), setting the correct physical dimensions, establishes the time scale of the process. The GME of Eq. (57) implying convolution in both space and time, makes it easy to evaluate the Fourier-Laplace transform of p(x, t), p(k, u), which reads [49]... 

This is simply the Fourier-Laplace transform of a Levy diffusion. The explicit expression for b coincides with the result afforded by the direct use of the GCLT [51]. 

Let us see if this is correct or not. In general, the Fourier-Laplace transform of Eq. (133) reads... 

On the fractal lattice, Dv is a constant, and holes exhibit the Gaussian characteristics, The self-similarity of the fractal has dilation symmetry shown in Eq. (8). Using the Fourier-Laplace transformation in time,... 

The theory is not limited in its application to the transient properties of amorphous polymers it can be used to make molecular interpretation and prediction of the dynamic viscoelastic properties of crosslinked polymers [24] as well. According to the Fourier-Laplace transformation, the complex tensile modulus can be separated into the real and imaginary parts... 

A wave may be viewed as a unit of the response of the system to applied input or disturbances. These responses could be in terms of physical deflections, pressure, velocity, vorticity, temperature etc., those physical properties relevant to the dynamics, showing up in general, as function of space and time. Any arbitrary function of space and time can be written in terms of Fourier-Laplace transform as given by,... 

For the purpose of linear analysis, we represent the perturbation quantities by their Fourier- Laplace transform via... 

The summation is over all the spanwise modes. One can use the above ansatz in three-dimensional Navier-Stokes equation and linearize the resultant equations after making a parallel flow approximation to get the following Orr-Sommerfeld equation for the Fourier- Laplace transform f of v ... 

The governing equation for the Fourier-Laplace transform is given by the following Orr-Sommerfeld equation,... 

In Fig. 4.13, the Fourier-Laplace transform of the solution at t = 788.5 is shown. In this case, the as miptotic deca3ung signal corresponds to Mode 2, while the effect of Mode 1 is not visible here, due to its high decay rate. The growing wave-front corresponds to the packet to the right of Mode 2. 

Here we are interested in the asymptotic behavior of the exact solution to Eq. (57) and we follow the analysis of Bologna et al. [52]. The most direct way to determine these properties is to take the Laplace transform in time and Fourier transform in space to obtain the Fourier-Laplace transform of the Liouville density... 

The three-term matrix recurrence relation, Eq. (208), may now be solved for the Fourier-Laplace transform C (oo) in terms of matrix continued fractions to yield [8]... 

The three-term recurrence Eq. (A2.2) can be solved exactly for the Fourier-Laplace transform f (m) in terms of ordinary continued fractions to yield... 

Instead of computing the correlation functions directly, one can take the Fourier-Laplace transforms, or spectral densities... 

The discussion presented below is most conveniently given in terms Fourier-Laplace transforms of the density fields... 

---