This assumes that the Fourier transform exists; i . e ., that the " j " ? axis is in the region of convergence of the Laplace transform.
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Conclusions can be drawn on the Region of Convergence based on Region of Support ( mathematics ) of the original sequence ( n _ 1, n _ 2 ).
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For example, the region of convergence of a power series is not necessarily an open ball; these regions are Reinhardt domains, the simplest example of which is a polydisk.
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That is, in the region of convergence " F " ( " s " ) can effectively be expressed as the absolutely convergent Laplace transform of some other function.
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There are several Paley Wiener theorems concerning the relationship between the decay properties of " f " and the properties of the Laplace transform within the region of convergence.
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Similarly, the set of values for which converges ( conditionally or absolutely ) is known as the region of conditional convergence, or simply the "'region of convergence "'( ROC ).
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In the frequency domain, the region of convergence must contain the unit circle ( i . e ., the locus satisfying | z | = 1 for complex " z " ).
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More generally, one can show that when c = 0, the interior of the region of absolute convergence is always a log-convex set in this sense . ) On the other hand, in the interior of this region of convergence one may differentiate and integrate under the series sign, just as one may with ordinary power series.
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