《电磁学》参考书籍英文电子书（Edward J. Rothwell Michael J. Cloud）Appendix-C

Appendix C Some Fourier transform pairs Note: G()=8(x)e-jkr dx, g(x) G()ekr dk, (x)+G(k). rect(x)+2 sinc k A(x)分Sm sgn(x)+2 eox2丌8(k-ko) 1分2r8(k) d s(x) 分(jk) (C.7) d"s(k) U(x)+π8(k)+ 8(k-nk0) (C.10) (C.11) eaU(x)分 ②2001 by CRC Press LLC

Appendix C Some Fourier transform pairs Note: G(k) = ∞ −∞ g(x)e− jkx dx, g(x) = 1 2π ∞ −∞ G(k)e jkx dk, g(x) ↔ G(k). rect(x) ↔ 2 sinc k (C.1) (x) ↔ sinc2 k 2 (C.2) sgn(x) ↔ 2 jk (C.3) e jk0 x ↔ 2πδ(k − k0) (C.4) δ(x) ↔ 1 (C.5) 1 ↔ 2πδ(k) (C.6) dnδ(x) dx n ↔ (jk) n (C.7) x n ↔ 2π j n dnδ(k) dkn (C.8) U(x) ↔ πδ(k) + 1 jk (C.9) ∞ n=−∞ δ  t − n 2π k0  ↔ k0 ∞ n=−∞ δ(k − nk0) (C.10) e−ax2 ↔ π a e− k2 4a (C.11) e−axU(x) ↔ 1 a + jk (C.12)

a+jk e cos bx U(x)分 a+jk)+b- (C.14) bxA (a +jk)+b cos kox <>[S(k + ko)+8(k-ko)] (C.16) sin kox j[8(k + ko)-s(- ko) (C.17) bx+h(sbx U(x) jk+b g(u)dr G(k) (C.19) a

e−a|x| ↔ 2a a2 + k2 (C.13) e−ax cos bx U(x) ↔ a + jk (a + jk)2 + b2 (C.14) e−ax sin bx U(x) ↔ b (a + jk)2 + b2 (C.15) cos k0x ↔ π[δ(k + k0) + δ(k − k0)] (C.16) sin k0x ↔ jπ[δ(k + k0) − δ(k − k0)] (C.17) 1 2 be− 1 2 bx  I0 1 2 bx + I1 1 2 bx U(x) ↔ jk + b jk − 1 (C.18) g(x) − ae−ax x −∞ eau g(u) du ↔ jk jk + a G(k) (C.19)