Appendix C Some Fourier transform pairs Note: G(k) = integraldisplay ∞ ?∞ g(x)e ?jkx dx, g(x) = 1 2π integraldisplay ∞ ?∞ G(k)e jkx dk, g(x) ? G(k). rect(x) ? 2 sinc k (C.1) Lambda1(x) ? sinc 2 k 2 (C.2) sgn(x) ? 2 jk (C.3) e jk 0 x ? 2πδ(k ? k 0 ) (C.4) δ(x) ? 1 (C.5) 1 ? 2πδ(k) (C.6) d n δ(x) dx n ? ( jk) n (C.7) x n ? 2π j n d n δ(k) dk n (C.8) U(x) ? πδ(k)+ 1 jk (C.9) ∞ summationdisplay n=?∞ δ parenleftbigg t ? n 2π k 0 parenrightbigg ? k 0 ∞ summationdisplay n=?∞ δ(k ? nk 0 ) (C.10) e ?ax 2 ? radicalbigg π a e ? k 2 4a (C.11) e ?ax U(x) ? 1 a + jk (C.12) e ?a|x| ? 2a a 2 + k 2 (C.13) e ?ax cos bx U(x) ? a + jk (a + jk) 2 + b 2 (C.14) e ?ax sin bx U(x) ? b (a + jk) 2 + b 2 (C.15) cos k 0 x ? π[δ(k + k 0 )+δ(k ? k 0 )] (C.16) sin k 0 x ? jπ[δ(k + k 0 )?δ(k ? k 0 )] (C.17) 1 2 be ? 1 2 bx bracketleftbigg I 0 parenleftbigg 1 2 bx parenrightbigg + I 1 parenleftbigg 1 2 bx parenrightbiggbracketrightbigg U(x) ? radicalBigg jk+ b jk ? 1 (C.18) g(x)? ae ?ax integraldisplay x ?∞ e au g(u) du ? jk jk+ a G(k) (C.19)