Page 272 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

5-42, Using Eqs. (5.161~) and (5.1551, find the Fourier transform of e-"Itl (a > 0).
From Eq. (5.155) we have

By Eq. (1.5) the even component of e-"'u(t) is given by

Thus, by Eq. (5.161~) we have

which is the same result obtained in Prob. 5.21 [Eq. (5.138)l.

5.43. Find the Fourier transform of a gaussian pulse signal

By definition (5.31)

Taking the derivative of both sides of Eq. (5.162) with respect to o, we have

Now, using the integration by parts formula

and letting
= -jwl and do = dr
we have

and