The Fourier Series Approximation

Another method of getting the numerical inverse is by the Fourier series approximation. The Laplace transform pairs are given as in Eqs. 9.1 and 9.3, written again for convenience 

We assume that fit) is piecewise continuous and of exponential order (Tq, that is, fit) I Mexp( Tot). 

The inverse equation (Eq. 9.2006) can be written in terms of the integral with respect to o as 

Following the procedure of Cnunp (1976), the approximation to the inverse is given by 

Numerically, the computation is done as follows (Crump 1976). Suppose the numerical value of fit) is needed for a range of t up to and the relative error is to be no greater than E. First T is chosen such that 

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Taking the Examples 9.16, 9.17, and 9.18, we compute the numerical inverse and show the results in the Tables 9.6, 9.7, and 9.8, where it is seen that the Fourier series approximation is a better approximation than the Zakian method. 

The Fourier series approximation is a better method to handle oscillating functions, but it requires more computation time than the Zakian method. With the advent of high speed personal computers, this is not regarded as a serious disadvantage. 

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