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
F(a) = j F(x)e—laxdx, F(x) = j F(a)eiaxda.
— TO —TO
By means of integration by parts, it is easy to find
d F 1 d F la
— = — —e—laxdx = — F (x)e—laxdx = laF (a).
dx 2n dx 2n
— TO —TO
F(x + X)e—laxdx = 2^ j F(n)e—lan+laXdn = elaXF(a).
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
f(oi) = 2П J f (t)elmtdt, f (t) = 2П J f (M)e—lmtdrn.
The inverse contour г is to be placed on the upper-half «-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.
By applying integration by parts to the integral below, it is straightforward to show
Let the initial conditions be
Ф, 0 < t < X 0, t < 0
f(t + X _ 21 f f (t + X)eilotdt _ 2^ j f (nV^-^dn
f(t – X _ eiaX f(co).