I)3 j Coinplrx A i ~ ~ l y sand i s the Inverse Laplacc Traiisforrn essential singulxities, but first, we have l o rmke soiiie restrictioiis We consiclcr adti makytic function F ( ) within a region, t1ia.t decays hy at lea~st,I / \ s \for sufficiently large v;iIuos of i s I (5.24) with Llie pohitive r e d nurrilm Ai To use thir; tlecrcasr tor large v d u e ~of / s / we choow tire circular integral around s o iis in Fig 5.6 rile malytic region contains all values of where Hc~{s} > c ~ ~Thc ~ path ~ i ol ~ intcgxation ~ consists of the path WrJ, which TUII:, paxallel t o tlir irnaginaiy axis at i j clistancc crp > o ~arid~t l ~ c~ path I1 , , whidi ii, a n arc with radius X aiouiid (lie oiigiii ~