LaPlace transforms tabulated

Inverse Laplace transforms have been tabulated for most analytical functions, including power, exponential, trigonometric, hyperbolic and other functions. In this context we require only the inverse Laplace transform which yields a simple exponential ... 

When we have found a solution for the Laplace transformed function, then we need to make an inverse transformation to find the solution in terms of time and coordinates. There are elegant techniques for doing this based on the theory of complex functions, but often these are not necessary since there exist extensive tables in mathematical handbooks of functions and their Laplace transformed functions. Only in cases where the relevant functions have not been tabulated will it be necessary to carry out the inverse transformation using these techniques. 

While mathematical insight is gained by use of Laplace transformations, Fourier transformation is used for gaining physical insight in terms of spectra. Theorems for Laplace transforms and the transforms of common functions are tabulated in the literature [Spil]. 

The integrals in Eq. [12] are not tabulated in widely used mathematical ref erences (Petit Bois, 1961, Abramowitz Stegun, 1965 Gradshteyn Ryzhik, 1980, p. 128-129). They appear to present an insurmountable difficulty to solution of the problem of describing the tracer concentration in a soil column due to a sinusoidal loading boundary concentration by applying the Laplace transform method with the readily applied convolution rule. 

In Table 7.1 the Laplace transforms of some basic functions have been tabulated. 

Integrals of e qmnential and trigonometric functions appear so frequently, they have become widely tabulated (Abramowitz and Stegun 1965). These functions also arise in the inversion process for Laplace transforms. The exponential, sine, and cosine integrals are defined according to the relations ... 

The previous example is very straightforward and could easily be inverted by use of partial fractions and a table of Laplace transforms. This next example may be tabulated. However, it is used here to demonstrate more clearly how the residue theorem may be useful for similar or more complicated inversions. Consider... 

Considerably less work exists addressing Leveque s problem for inlet channel flow with wall reaction. Carslaw and Jaeger [58] and Petersen [68] presented solutions for plug-flow conditions, using the Laplace transform. Pancharatnam and Homsy [69] used the same technique for laminar flow. The inversion of the transformed solution is given in terms of an infinite summation with coefficients given by recurrence relations (first 24 out of 50 coefficients are tabulated). Ghez [70] considered a first-order reversible reaction with the same solution method. Moreover, asymptotic expansions in the limits of fast and slow reactions were presented. 

The Laplace transform of many functions have been evaluated and the results tabulated. Extensive tables may be found in many texts on the subject. The Laplace transform of a few functions is given in the table below. 

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