Laplace transform factorization

Laplace transforms 279-286 mapping of a function 271 integrating factor 56-59, 86, 360 integration 43-61 along a curve 51-52 by parts 46 by substitution 45-46 chemical kinetics 47-48 constant of integration 43,49, 89, 124... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

The presence of the h(z, P) factor makes Eq. (7.44) different from a Laplace transform of C(z). If the z dependence of h(z, P) is ignored,(34 36) then calculated concentrations of fluorophore near an interface derived from collected fluorescence are approximations. Also, the P dependence in the tf1,11 causes the integral in Eq. (7.44) to differ from the form of a Laplace transform even after the excitation term is factored out. 

If the excitation electric field is an s-polanzed evanescent field instead of the above p-polarized example, then wH 11 [ = wHJI(z)] does not depend upon p. Therefore, an approximate C(z) can be calculated from the observed fluorescence (P) (obtained experimentally by varying 0) by ignoring the z dependence in the bracketed term in Eq. (7.45) and by inverse Laplace transforming Eq. (7.44) after the ,(0, /J) 2 term has been factored out.,37 39)... 

B. PARTIAL-FRACTIONS EXPANSION. The linearity theorem [Eq. (18.36)] permits us to expand the function into a sum of simple terms and invert each individually. This is completely analogous to Laplace-transformation inversion. Let F, be a ratio of polynomials in z, Mth-order in the numerator and iVth-order in the denominator. We factor the denominator into its N roots pi, P2, Ps,... 

Fig. 2.49 Dynamic structure factor at two temperatures for a nearly symmetric PEP-PDMS diblock (V = 1110) determined using dynamic light scattering in the VV geometry at a fixed wavevector q = 2.5 X 10s cm-1 (Anastasiadis et al. 1993a). The inverse Laplace transform of the correlation function for the 90 °C data is shown in the inset. Three dynamic modes (cluster, heterogeneity and internal) are evident with increasing relaxation times.

Laplace transforms, introducing the conjugate variables rm, rn = 1,..., M (In field theory a variable of the type rfn would be addressed as a mass1.) After the Laplace transform a propagator of momentum k in the m-th polymer line yields a factor G7jF(k rm). All segment integrations are eliminated. 

General considerations of renormaiizability are best carried through within the field theoretic formulation. We recall from Sect. 7.2 that a Laplace transform with respect to the chain length variables eliminates the segment summations. In the resulting field theoretic formulation a propagator line of momentum k, being part of the m-th polymer line, yields a factor (cf, Eq. (7.17))... 

Note that in the limit q —> 0 of vanishing latency time, we obtain w(s) —> fo(s), as is to be expected. The only remaining problem then is to extract the waiting time distribution w(t) from its Laplace transform u>(s), a task complicated by the appearance of exponential factors in g< in the denominator in Eq. (22). 

Solve the system of algebraic equations in the frequency domain to obtain the transfer function between the input and sampling sites. The Laplace transform of the probability that a molecule survives in the sampling time after time f is this transfer function where substitution of the multiplying fi (s) factor in the sampling site by the corresponding Si (s) was made. 

Jung et al. [81] reported a variant of this approach that used the multiplicative factor of zero for the SS component and 1.3 for the OS component. This calculation, SOS-MP2 (scaled opposite-spin MP2), can be performed with only an 0(n4) operation cost when combined with Almlof s Laplace transform technique [82], The SOS approximation can be applied to CIS(D) [69], A similar simplification was often adopted in the GW method under the name COHSEX approximation [32] also partly from an operation cost consideration. 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

Taking the Fourier-Laplace transform of Eq. (9), the dynamic structure factor can be obtained as,... 

It is possible to use the idea of evaluating the polynomial at a real or complex value as an aid to proving that one polynomial is a factor of another, by showing that all the roots of the first are also roots of the second. In fact in the Laplace transform the domain is definitely that of complex numbers, and [CDM91] uses this interpretation very fluently and to good effect. 

Typically when linear partial differential equations are solved using the Laplace transform method the solution obtained in the Laplace domain can be represented as in equation (8.7) and q(s) usually has an infinite number of roots. If s = p . n = 1..00 are the distinct roots of q(s), q(s) can be factorized as... 

Let us calculate 3t(0, r S) up to a numerical factor by performing the inverse Laplace 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. 

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transforms. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization technique. This method has been generalized to charged hard spheres. 

To solve (13.73) with the initial condition given by (13.77), we can either use an integrating factor and an assumed form of the solution or attack it directly with the Laplace transformation. Both approaches are illustrated in Appendix 13.2. The desired solution is then... 

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