Inverting the Laplace Transform

The function f is called the convolution cfh and g and the integrals (Equation 3.95) are called convolution integrals. 

Essentially, the convolution integrals do not have all the properties of ordinary multiplication. 

This example illustrates inversion using the convolution integral. Consider 

The third standard method of inverting a Laplace transform is by making use of the Residue theorem [6,15,16,18,22]. The transform function F(s) is analytic, except for singularities. In this discussion, when F s) is analytic, the inverse transform of F(s) is given by 

The following example illustrates the use of the residue theorem and should serve to clarify certain new terminology  

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By applying the Laplace transform to the U-series decay equation, one obtains simple linear equations that can be solved for the Laplace transforms of Ni (the number of nuclei i in the system). By inverting the Laplace transforms using tables, the time-dependent solutions are directly obtained. The Laplace transform for Equation (1) is ... 

The Laplace transform is similar to a one-sided Fourier transform, except that it has a real exponential instead of the complex exponential of the Fourier transform. If we consider complex values of the variables, the two transforms become different versions of the same transform, and their properties are related. The integral that is carried out to invert the Laplace transform is carried out in the complex plane, and we do not discuss it. Fortunately, it is often possible to apply Laplace transforms without carrying out such an integral. We will discuss the use of Laplace transforms in solving differential equations in Chapter 8. 

After transforming the equation into the Laplace domain and solving for output variables in term of s, we can transform back in the time domain by inverting the Laplace transform ... 

The maximum entropy method can handle Laplace transforms such as those found in pulse fluorometry, without restricting the validity of the solution or suffering from any instability. It allows the recovery of the distribution of exponentials describing the decay of the fluorescence (i.e., inverting the Laplace transform), which is, in turn, convolved by the shape of the excitation flash. Also, it can determine the background level and amount of parasitically scattered radiation. 

Using the linear solubility relationship (Eqn (11)) to link the interfacial concentration and temperature, C (t) can be deduced by inverting the Laplace transformed Eqn (12), for a constant value of the reaction rate constant (ie assuming zero activation energy for the reaction). The algebraic... 

Inverting the Laplace transform gives the simple exponential decay... 

The third step is to invert the Laplace transform or find the inverse transform of v(x, s) with the aid of a table [7] to get... 

The limiting behavior of the correlations at large distances is given by an Omstein-Zemike form (introduced in Section X.1.4). Inverting the Laplace transformation [eq. (X.39)], it is then possible to fmd the asymptotic form of or of [Pg.279]

The data analysis consists of inverting the Laplace transformation Eq. (6). However, the measured has noise and is bandwidth limited... 

In a second class, a density expansion is used to obtain a series approximation for G (e) that is accurate at short times and small ncent rat ions. One approach is to construct a Fade approximant for G (e) from this series that causes G (t) to decay to zero at long times. A second approach is to construct cumulant approximants for the series by inverting the Laplace transform of the density exMnsion and re-expressing the series as a short-time expansion for fn[G (t)]. 

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