Inverse of the Laplace Transformation

If the Laplace transform of a function /(/) is f s), then f(t) is the inverse Laplace transform of f(s). Although an integral inversion formula can be used to obtain the inverse Laplace transform, in most cases it proves to be too complicated. Instead, a transform table (1), is used to find the image function f f). For more complicated functions, approximate methods are available. In many cases the inverse of a ratio of two polynomials must be 

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Inversion of the Laplace transform for a step change in Cq gives the analytical solution derived previously. 

The inversion of the Laplace transform presents a more difficult problem. From a fundamental point of view the inverse of a given Laplace transform is known as the Bromwich integral. Its evaluation is carried out by application... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

J. G. McWhirter and E. R. Pike, On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A Math. Gen. 11, 1729-1745 (1978). 

Inversion of the Laplace transformation gives the time function. If we have a transfer function the unit step function is C [G(,y s] and the impulse response is C" [G(,)]. 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

The cosech function are expanded as a power series in kT. Inversion of the Laplace transform then gives a series solution for the density of states, whose first term is the same as the Marcus-Rice correction, Eq. (85), and whose subsequent terms are corrections containing smaller powers of -f- E. Ultimately the expansion of the partition function gives terms with negative powers of kT, which invert to delta functions and derivatives of delta functions, but truncating the series before this happens gives good smooth approximations to the density of states. 

Maple cannot find the inverse of the Laplace transform ... 

Determination of the distribution of modes y4(F) and the related distribution of sizes requires inversion of the Laplace transform, which is an ill-defined problem for a limited data set containing any noise. There are some numerical programs (such as CONTIN that attempt to perform this inverse transformation. The resulting distributions do sometimes (but not always) correlate (but not coincide) with the actual distribution of hydrodynamic radii in solution. 

As pointed out above, the critical point in finding the solution to a differential equation using Laplace transforms is the inversion of the Laplace transforms. In this section we will study a method developed by Heaviside for the inversion of Laplace transforms known as Heaviside or partial-fractions expansion. 

Inversion of the Laplace transform is then readily done using the specific... 

The inversion of the Laplace transforms of Eq. (9.105) with the initial conditions of Eq. (9.102) gives the availability... 

These are the Cauchy-Riemann conditions, and when they are satisfied, the derivative dw/ds becomes a unique single-valued function, which can be used in the solution of applied mathematical problems. Thus, the continuity property of a complex variable derivative has two parts, rather than the one customary in real variables. Analytic behavior at a point is called regular, to distinguish from nonanalytic behavior, which is called singular . Thus, points wherein analyticity breaks down are referred to as singularities. Singularities are not necessarily bad, and in fact their occurrence will be exploited in order to effect a positive outcome (e.g., the inversion of the Laplace transform ). 

The expression (4.3.23) can also be considered for direct inversion of the Laplace transform of the function (5, T) by expanding it in Taylor series about T = 0... 

Ross, D. A., Dhadwal, H. S., Regularized Inversion of the Laplace Transform Accuracy of Analytical and Discrete Inversion, Part. Part. Syst. Charact, 1991, 8, 282-286. 

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