Ma200

Chapter 11

Laplace Transforms

Reading There is no speci…c essential reading for this chapter. It is essential that you do some reading, but the topics discussed in this chapter are adequately covered in so many texts on the ‘ applications of calculus’and in texts dedicated to this important subject that it would be arti…cial and unnecessarily limiting to specify precise passages from precise texts. The list below gives examples of relevant reading. (For full publication details, see Chapter 1.) Ostaszewski, A., Advanced Mathematical Methods, Chapter 21. Spiegel, M.R., Laplace Transforms, Schaum Outlines Series, McGraw-Hill, 1965. Spiegel, M.R., Advanced Calculus, Schaum Outlines Series, McGraw-Hill, 1974.

Introduction This chapter summarizes and establishes properties of the Laplace transform which have already been discussed in earlier chapters, and then develops further properties. Examples of applications given here include solutions of second order di¤erential equations with non-constant coe¢ cients and certain probability distributions that take the form of Laplace transforms. An earlier application in Section 9.3 solved a partial di¤erential equation. The main new tool of this chapter is the convolution theorem. Convolutions were …rst considered in Section 9.7.1.

11.1

Learning objectives

At the end of this chapter and the relevant readings, you should be able to: de…ne the Laplace Transform for functions of at most exponential growth recall the basic transforms, namely those of: e

at

; t ; cos at; sin at

recall the shift and scaling rules and the rules for computing L(f 0 ) and L(t f (t)) apply these rules to solve di¤erential equations quote the Laplace transform of a convolution of two functions and be able to use it to …nd inverse Laplace transforms deduce the formula relating the Beta and Gamma functions use the Convolution Theorem.

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11.2

Recapitulation of the properties...