Inversion of Laplace Transforms by Contour Integration

At the beginning of this chapter, we quoted the Mellin-Fourier Inversion theorem for Laplace transforms, worth repeating here 

It may now be clear why the factor 1 /Ivi appears in the denominator of the Inversion theorem. It should also be clear that 

When this is done, the reader can see that Laplace inversion is formally equivalent to contour integration in the complex plane. We shall see that exceptional behavior arises occasionally (singularities owing to multivaluedness, for example) and these special cases will be treated in the sections to follow. Our primary efforts will be directed toward the usual case, that is, pole and multiple pole singularities occurring in the Laplace transform function Fis). 

We shall first consider the Inversion theorem for pole singularities only. The complex function of interest will be fis) = e F(s). The contour curve, denoting selected values of s, is called the First Bromwich path and is shown in Fig. 9.4. 

The real constant is selected (symbolically) to be greater than the real part of any pole existing in the denominator of F(s). Thus, all poles of F(s) are to the left of the line labelled y4B (Fig. 9.4). It is clear that the semicircle BCDEA can become arbitrarily large in the limit as I - , thereby enclosing all possible poles within the region to the left of line AB. 

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