# The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function

## Abstract

We now admit singularities of *F*(*s*) to the left of *a* which are not all single-valued. In the case that the first encountered singularities to the left of *a* are single-valued, one can employ the previous method and thus separate from *f*(*t*) the corresponding residues, until one finally encounters a many-valued singularity. Thus, we may, without loss of generality, assume that in the first encountered singular point to the left of *a*, that is *the singular point α*_{ 0 } *with largest real part* < *a*, *F*(*s*) *has a many-valued singularity*, perhaps of the character (*s* − *α*_{ 0 })^{1/2} or (*s* − *α*_{0})^{−1/2} or log (*s* − *α*_{0}) or (*s* − *α*_{ 0 })^{1/2} log (*s* − *α*_{ 0 }) etc. Possibly, one may encounter more than one singular point with identical largest real part *< a*; this particular situation will be discussed at the end of this Chapter.

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