Transformation inverse

2 Inverse Transformation The normal coordinate is given as a linear combination of the bead positions. Conversely, we can express r as a linear combination of q, (i = 1, 2.N). FromEq. 3.118, we find 

Note that the superposition coefficient for i = 0 and is a half of the others. For n 

In practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs, as given in Table 3.1. 

---

Cable J R and Albrecht A C 1986 The inverse transform in resonance Raman scattering Conf. sponsored by the University of Oregon ed W L Peticolas and B Hudson... 

Cable J R and Albrecht A C 1986 A direct inverse transform for resonance Raman scattering J. Chem. Phys. 84 4745-54... 

Joo T and Albrecht A C 1993 Inverse transform in resonance Raman scattering an iterative approach J. Phys. Chem. 97 1262-4... 

Lee S-Y 1998 Forward and inverse transforms between the absorption lineshape and Raman excitation profiles XVith int. Conf on Raman Spectroscopy ed A M Heyns (New York Wiley) pp 48-51... 

The function L[f t)] = g(s) is called the direct transform, and L [g. s)] =f t) is called the inverse transform. Both the direct and the inverse transforms are tabulated for many often-occurring functions. In general,... 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

The inverse transform must he found, usually from a table of inverse transforms. 

A = sampling rate (e.g., number of samples per second) The Fourier transform and inverse transform are... 

This teehnique transforms the problem from the time (or t) domain to the Laplaee (or. v) domain. The advantage in doing this is that eomplex time domain differential equations beeome relatively simple. v domain algebraie equations. When a suitable solution is arrived at, it is inverse transformed baek to the time domain. The proeess is shown in Figure 3.2. 

Equation (3.25) is in the form given in Faplaee transform pair 5, Table 3.1, so the inverse transform beeomes... 

We now take the inverse transform, which converts Ca to Ca- From transform No. 4 in Table 3-1, with a = — fc, we obtain... 

By taking inverse transforms of the separate terms, using Nos. 1 and 4 of Table 3-1, we find y ... 

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

The transform Ca can be found by alternative algebraic routes, and it will appear to be different from Eq. (3-89), and the inverse transform will not appear to be identical to Eq. (3-90), but these differences in appearance result because the parameters are composite quantities.]... 

Applying the general partial fraction theorem, Eq. (3-72), to Eq. (3-96) and then taking inverse transforms gives Eq. (3-99). 

Solutions derived by Laplace transformation are in terms of the complex variable s. In some cases, it is necessary to retransform the solution in terms of time, performing an inverse transformation... 

Just as there is only one direct transform F(s) for any f(t), there is only one inverse transform f(t) for any F(s) and inverse transforms are generally determined through use of tables. 

Consequently, the tedious and usually impossible evaluation of the inverse transform of the transformed partial differential equations (236) becomes an unnecessary procedure, and a significant mathematical simplification results. 

The copper EXAFS of the ruthenium-copper clusters might be expected to differ substantially from the copper EXAFS of a copper on silica catalyst, since the copper atoms have very different environments. This expectation is indeed borne out by experiment, as shown in Figure 2 by the plots of the function K x(K) vs. K at 100 K for the extended fine structure beyond the copper K edge for the ruthenium-copper catalyst and a copper on silica reference catalyst ( ). The difference is also evident from the Fourier transforms and first coordination shell inverse transforms in the middle and right-hand sections of Figure 2. The inverse transforms were taken over the range of distances 1.7 to 3.1A to isolate the contribution to EXAFS arising from the first coordination shell of metal atoms about a copper absorber atom. This shell consists of copper atoms alone in the copper catalyst and of both copper and ruthenium atoms in the ruthenium-copper catalyst. 

Figure 2. Normalized EXAFS data (copper K absorption edge) at 100°K, with associated Fourier transforms and inverse transforms, for silica supported copper and ruthenium-copper catalysts. Reproduced with permission from Ref. 8. Copyright 1980, American Institute of Physics.

---