Like the Fourier transform, the Laplace transform is used in a variety of applications. Perhaps the most common usage of the Laplace transform is in the solution of initial value problems. However, there are other situations for which the properties of the Laplace transform are also very useful, such as in the evaluation of certain integrals and in the solution of certain integral equations. In this chapter we will briefly discuss applications of the Laplace transform in all of the above named areas.
An interesting application of Laplace transforms involves the evaluation of certain integrals, particularly those containing a free parameter. In some cases we simply recognize the integral as a special case of a Laplace transform for a particular value of the transform variable p. Other integrals may be solved by first taking the Laplace transform of the integrand with respect to a free parameter (not the variable of integration). The resulting integral is hopefully easier to evaluate than the original, and by applying the inverse Laplace transform we obtain our desired result. This latter procedure is direct and often simple, but it requires the interchange of two limit operations, so some caution should be exercised in its usage.
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