Table of Laplace Transformations

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. 

Thus the Laplace transformation constitutes a method of integration, and a table of Laplace transforms plays a role in this process that is analogous to a table of... 

Many other Laplace transforms can be derived in this way. Extensive tables of Laplace transforms are available and are of routine use, particularly by electronics engineers. 

Tables of Laplace transforms for a large number of functions have been calculated, and can be obtained from published data. In the present example, the transformed equation is...

From a table of Laplace transforms (Jenson and Jeffreys, 1963, p. 532),... 

Inversion with a table of Laplace transforms gives the results, A = Aq expC k1t) (12)... 

The following is a short table of Laplace transforms for quick reference. 

Hence, from Table of Laplace transforms (Appendix 7.1) ... 

We shall now determine the inverse Laplace transform of this function to find the concentration of B at time t. In tables of Laplace transforms the following pair has been found ... 

These properties allow one to solve the integration problem in transform space and then back-transform the result into real space. There are extensive tables of Laplace transforms, some of which are given in Table 2.9. 

At this point, the inverse Laplace transformation of equation (6.5.19) is carried out. In order to do this, one must consult a table of Laplace transforms such as the one given in appendix A. Operating on each term individually, one obtains... 

The final step is to perform inverse Laplace transformation on c x, s) to obtain c x, t). Using the table of Laplace transformations, inverse transformation of equation (6.5.27) results in... 

The second-order partial differential equations given in the text of the paper contain time derivatives of concentration (dC/dt) and terms containing dC/dz and C. The solutions of these equations, unless referred to a literature source, were obtained by the method of Laplace transformation with the aid of standard tables of Laplace transforms. Good working summaries of the Laplace transformation method as applied to... 

Oberhettinger, F., and L. Badii. 1973. Tables of Laplace transforms. Springer-Verlag, New York. Oppenheimer, S.F., D.D. Adrian, and A.N. Alshawabkeh. 1999. A river water quality model for time varying BOD discharge concentration. Math. Probl. Eng. 5 193-221. 

However, to see how the transform technique works, it is instructive to apply it to several simple examples that will also be needed later. Many of these basic relationships are listed in tables of Laplace transforms. 

G. E. Roberts and H. Kaufman, Table of Laplace Transforms, Saunders, Philadelpia, 1966. 

R. V. Churchill, Operational Mathematics, 3rd edn., McGraw-Hill, New York, 1972 G. E. Roberts and H. Kaufman, Tables of Laplace Transforms, Saunders, Philadelphia, 1966. 

Table of Laplace Transforms 673 Table of Laplace Transforms (Continued)... 

Oberhettinger, R, Badii, L., Tables of Laplace Transforms, Springer-Verlag, New York, 1973. 

The differential equation was solved only by transformation, which can be done using Laplace transform tables, the solution of an algebraic equation, and the inverse Laplace transform using tables. Tables of Laplace transforms can be easily... 

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