The_Cambridge_Handbook_of_Physics_Formulas

54 Mathematics
Fourier transform pairs a f(x) ⇀↽ F(s)=
f(x)=
∫ ∞
−∞
f(x)e −2πisx dx (2.499)
f(ax) ⇀↽ 1 F(s/a)
|a|
(a ≠ 0, real) (2.500)
f(x−a) ⇀↽ e −2πias F(s) (a real) (2.501)
d n
dx n f(x) ⇀ ↽ (2πis) n F(s) (2.502)
δ(x) ⇀↽ 1 (2.503)
δ(x−a) ⇀↽ e −2πias (2.504)
e −a|x| ⇀↽
2a
a 2 +4π 2 s 2 (a>0) (2.505)
xe −a|x| ⇀↽
8iπas
(a 2 +4π 2 s 2 ) 2 (a>0) (2.506)
e −x2 /a 2 ⇀↽ a √ πe −π2 a 2 s 2 (2.507)
sinax ⇀↽ 1 [ (
δ s− a ) (
−δ s+ a )]
(2.508)
2i 2π 2π
cosax ⇀↽ 1 [ (
δ s− a ) (
+δ s+ a )]
(2.509)
2 2π 2π
∞∑
δ(x−ma) ⇀↽ 1 ∞∑ (
δ s− n )
a
a
(2.510)
m=−∞
{
0 x0
{
1 |x|≤a
f(x)=
0 |x| >a
⎧(
⎨
1− |x| )
|x|≤a
f(x)= a
⎩
0 |x| >a
n=−∞
(“step”) ⇀↽ 1 2 δ(s)− i
2πs
(“top hat”) ⇀↽ sin2πas
πs
(“triangle”)
⇀↽
(2.511)
=2asinc2as (2.512)
1
2π 2 as 2 (1−cos2πas)=asinc2 as (2.513)
a Equation (2.499) defines the Fourier transform used for these pairs. Note that sincx ≡ (sinπx)/(πx).