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Foundation Mathematics for the Physical Sciences by K. F. Riley, M. P. Hobson

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13

Laplace transforms

In Chapter 6 we discussed how complicated functions f(x) may be expressed as power series. Although they were not presented as such, the power series could all be viewed as linear superpositions of the monomial basic set of functions, namely 1, x, x2, x3, . . . , xn, . . . Natural though this set may seem, they are in many ways far from ideal: for example they possess no mutual orthogonality properties, a characteristic that is generally of great value when it comes to determining, for any particular function, the multiplying constant for each basic function in the sum. Moreover, this particular set of basic functions can only be used to represent continuous functions.

In the case of original functions f(t) that are periodic, ...

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