Laplace integral

The Laplace transformation is based upon the Laplace integral which transforms a differential equation expressed in terms of time to an equation expressed in terms of a complex variable a + jco. The new equation may be manipulated algebraically to solve for the desired quantity as an explicit function of the complex variable. 

F(s) = the Laplace transform of f, expressed in s, resulting from operating on f(t) with the Laplace integral. 

When the number of hydrocarbon groupings is sufficiently high, one could rank the hydrocarbon groupings according to the kinetic constant, and replace the sum in Eq. (16) by a Laplace integral... 

Here erfc( ) is the error function complement, a mathematical function of the argument u given by 1 - erf( ), where erf( ) is the error or Euler-Laplace integral. This integral in turn is defined by the expression... 

This difficulty is somewhat hidden in the usual presentation of the subject.10 One starts at t = 0 with a solution un° of the unperturbed equation (Eq. (14)) and one switches in the interaction XV. The solution is then represented for t 0 by a Laplace integral... 

There are numerous useful integral transforms, each of which is specified by a two-variable function Kit, p) called the Kernel function or nucleus of the transform. The Laplace transform or the Laplace integral of a function fit), defined for... 

Figure 2.16 The strip of convergence for the Foiu"ier-Laplace integral in the wave number plane...

Vander Pol, B. and Bremmer, H. (1959). Operational Calculus Based on Two-Sided Laplace Integral, Cambridge University Press, U.K. 

The Laplace integral transformation, used in Section 4.3.1, allows the indentifica-tion of its kernel as K(z,p) = K(z,s) = e . It corresponds to the case when we produce a transformation with time. So, for this case, we particularize the relation (4.151) as ... 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

The system equation along with boundary and initial conditions were solved analytically using a Laplace integral transform (1.) a computer program by Cleary (14) was modified for use in this study. 

Nevertheless, there is a strong similarity between the exponentials of Eqs. (2.53) and (2.22). To proceed, it is worth noting, that the essential part of the Laplace-transform which allows the calculation of the multiple integral for Gl, see Appendix A, is determined by a stationary point of the Laplace-integral only and hence (within this approach) the same result is obtained using Gwi instead of G. Thus, following the same steps as presented in Appendix A, the essential part of the free energy of the amorphous fraction can be written as... 

A more complete study of A shows that the Hille-Yosida theorem [8] applies and does indeed insure the existence of a semi-group of bounded operators T(t) providing the unique solution to the problem posed. The solution may be represented in the form of a Laplace integral. 

For calculations one can use the quantitative values < (A)=/o presented in Table 8.1. However, in the case of d < 4 space, an approximated analytical expression /o can be formulated, using the Laplace method [11]. With this aim we have rewritten (8.32) as a d-multipled Laplace integral... 

Another Unear-scaUng MP2 algorithm was proposed in [129] that is based on the atomic-orbitals Laplace-transform (LT) MP2 method [137]. In this method, the energy denominators (5.32) in (5.30) are ehminated by Laplace-transformation, which paves the way to express the MP2 energy directly in the basis. The price to pay is the additional Laplace integration, which is carried out by quadrature over a few (8-10) points. For each of the quadrature points an integral transformation has to... 

It can be transformed into a Laplace integral or Laplace transform by putting 1/t = s, the relaxation frequency, and introducing a frequency function N s)ds defined as... 

The importance of this representation is that when G(r) has been determined, the relaxation time spectrum can be found, in principle, by standard methods for the inversion of the Laplace integral. In practice this requires computation methods, as it is not usually possible to find an analytical expression to fit the stress relaxation modulus. 

Creep compliance. In this case the rate of change of creep compliance is expressed as a Laplace integral. Thus... 

There are three common routes one can take to obtain the decay characteristics of the particles motions from the ACF. The first is to calculate the mean value and other moments ignoring the details ofq(F). The second is to use an assumed or known analytical distribution form. In this case, the task is simply to adjust the variables of the analytical form so a best fit to the experimental data can be found. The third route is to resolve the complete distribution by performing an inversion of the Laplace integral equation (Eq. 5.7). Detailed mathematical derivations of these methods can be found in the literature, but are beyond the scope of present text. The important aspect for most readers is to understand the nature, jargon (technical terms), the application range, and the most appropriate use of each method. A detailed discussion of PCS data analysis methods can be found elsewhere [23]. 

Inversion of Laplace Integral Equation Judgment of the Computed Distribution Data Analysis ofMultiangle Measurement... 

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