Fourier and Laplace Transforms
A.1 Fourier Transform
Let f(x) be an absolutely integratable function; i. e., f—°TO F(x)dx < to, then F(x) and its Fourier transform F(a) are related by
TO to
F(a) = j F(x)e—laxdx, F(x) = j F(a)eiaxda.
— TO —TO
Derivative Theorem
By means of integration by parts, it is easy to find
TO TO
d F 1 d F la
— = — —e—laxdx = — F (x)e—laxdx = laF (a).
dx 2n dx 2n
— TO —TO
Shifting Theorem
TO
F(x + X)e—laxdx = 2^ j F(n)e—lan+laXdn = elaXF(a).
— TO
A.2 Laplace Transform
Let f(t) satisfy the boundedness condition /TO e—ct f (t)dt < to for some c > 0, then f(t) and its Laplace transform f(^) are related by
TO
f(oi) = 2П J f (t)elmtdt, f (t) = 2П J f (M)e—lmtdrn.
0 г
The inverse contour г is to be placed on the upperhalf «plane above all poles and singularities of the integrand (see Figure A1). This condition is necessary in order to satisfy causality.
Figure Al. Complex «plane showing the inverse contour Г, • poles, ~~~~ branch cut.
Derivative Theorem
By applying integration by parts to the integral below, it is straightforward to show
Shifting Theorem
Let the initial conditions be
Ф, 0 < t < X 0, t < 0
then,
о о f(t + X _ 21 f f (t + X)eilotdt _ 2^ j f (nV^^dn

Similarly,
f(t – X _ eiaX f(co).