1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.4 • THE FOURIER TRANSFORM 539

Table 12.1 gives some important properties of the Fourier transform.

Linearity

Symmetry

Time scaling

T ime shifting

Frequency shifting

Time differentiation

Frequency differentiation

Moment theorem

i (aU1 (t) + bU2 (t)) = ai(u1 (t)) + bJ(U2 (t))

If i(u (t)) = F (w), then i(F (t)) = i,.U (-w).

J (U (at))= ~F (!£)

J (U (t - to))= e-itow F (w)

J(e-iwotu(t)) =F(w-wo)

i (U' (t) ) = iwF(w)

d"fw~w) = i((-it)" U (t))

If M n = J~ 00 tnU (t) dt, then (-i)" M,. = 21f p(n) (0).

Table 12.1 Properties of the Fourier Transform

1

• EXAMPLE 12.5 Show that F (e-ltl) = ( 2 ).

1f 1 +w

Solution Using Equation (12- 25), we obtain

= 1 1° e(l-iw)tdt + 1 1"° e<- I - iw)tdt

21f -oo 21f 0

= 1. e<I-iw)t + 1. e(- 1- iw)t

l

t=O 1 t=oo

21f (1 -iw) t =-oo 21f (- 1 -iw) t=O

1 1 1

= 21f(l-iw) + 21f(l+iw) = 1f(l+w2)'

establishing the result.

•EXAMPLE 1 2. 6 Show that J (~ 2 ) = ~e-lwl.

1 + t 2