E-transforms
F. Ragab · Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics · 1967
Th e fo ll o wing tra ns form pair is es tabli s hed: ;:'1x) = JX (xy),'[Ep ; a,•: q: P,: CrY) ~"lj(y) dy." o q: I -PI+ -: (xy) ' " II ~ / . 1 J j(x) = II " {x (xy) -kE j " I .I 1; +-1) + I : 1 -<1, +-, I ;:,1y) dy.II n wh e re II. is an y pos itive integer a nd E is Mac Ro be rt's fun ction a nd th e ge ne ra li zed Ma c Ro be rt's fun cti o n, res pec t ive ly.S pec ia l c ho ices of t.h e para me t.e rs in th e las t tra ns form lead in turn to th e de riva ti o n of Ha n ke l tra ns form , V-tra ns form , K-tra ns form , Fo uri e r tran s fo rm , Lap lace tra ns fo rm a nd ot.h e r int egra l tran sform s with ta bl es to illu strate t.h ese ne w tra ns form s. Key Wo rd s : Fo uri e r-tra nsform , fun c tiona l tra ns form s, ge ne ra lized Mac Ro bert's fun c ti o n, Hank e l tra ns fo rm , K-tra ns fo rm , L a pl ace tra ns fo rm , M ac Ro bert ' s fun c ti o n, V-tra ns form .