and the s i cp fuiict ion ~ ( t ) \lit\ will deal wit,h the evaluation of tlic Foiirier tmniornis iii thc coursc of this chapter The Fourier tr;tnsform of the s k p fimction contains a delta impitlse 6(d).Up to iiow, TW ltax7e encounimeti it in the time-tlorriain, biit not in the ~requenc.y-dorrii~in As Laplace trimsforms in the region of corivcrgeiice are analyt ic fimctions there is no room there Erw gmrrafiscd functions Ro-,wver, with Fourier traxisionns that arc fiuietions of a real parameter w (jiist as with fimci ioris of tirtci), thc extension to cover gewra1isc.d functions does make smse T h e popularity of t hc Fouricr transform conies from the posqiblity of assigning a spcctnin7 with the help of the dcltit irripiilsc to riiariy iniporkd prattica.21functions for which hew is no Laplacc transiorm Sonir eltmcntctry examples are nts ( ~ g r ( t ) = 1) arid t,he trigovtornctric frmctions siriwi,t and cob wet ities a ee s rie S it can bc scen immediately that llie FoinieI transform of a function of tiinc is the axis s = j w of the coniplex plane sitirit a5 i t s Lixplace transtoriii on the imagina That is of coiiise only triie if the r,aplacc transiorm cxists as well T h e coimcct,iou bctween Laplace and Fourier t ransl'orrns is: if the region of coriwrgerie