Bromwich path

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. 

Figure 9.4 First Bromwich path for pole singularities.

If we insure that the real part of s is always greater than a (as we have seen in the previous section regarding the first Bromwich path), then the upper limit is zero and... 

Inversion Theory for Multivalued Functions The Second Bromwich Path 379... 

Figure 9.13 Contour (second Bromwich path), which does not enclose branch point at s 0.

Thus, if any contour is drawn so that the branch point is encircled, then multivalued behavior arises. The principle to ensure analyticity is simple branch points cannot be encircled. There is considerable range and scope for choosing branch cuts to ensure analytic behavior. Now, if no other singularities exist in the contour selected, then Cauchy s First Integral theorem is valid, and we denote the new contour C2 as the second Bromwich path, Br2 hence. 

We shall use the second Bromwich path drawn in Fig. 9.13, where the branch point is located at 5 = 0 and the branch cut extends to infinity. We write first that, since no other singularities exist... 

The arguments put forth earlier regarding integrals around the first Bromwich path are still applicable to parts of the Brj curve (e.g., Eqs. 9.92-9.94), so when i - 00, we can write immediately... 

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