Find the following Fourier sine and cosine transforms or inv

Find the following Fourier sine and cosine transforms or inverse Fourier transforms by your favorite method. (my favorite method is the tables) a) F_c^-1 (1/1+w^2) b) F_c^-1 (e^-w^2) c) F_s (e^-pi x) d) F_s (x^-1/2)

Solution

a] By using standard result of fourier cosine transform

F_c {e^-ax } = a/(a² + w²)

for a=1

  F_c {e^-x } = 1/(1+ w²)

  F_c^-1 {   1/(1+ w²) } = e^-x

[b]    F_c {e^-ax2 } = 1/2*sqrt(3.14/a)*e^-w2/4a

   for a=1/4

  F_c {e^-x2/4 } = 1/2*sqrt(3.14*4)*e^-w2 = sqrt(3.14) * e^-w2

F_c^-1 { e^-w2 } = 1/sqrt(3.14) *e^-x2/4

[c]

By using standard result of fourier sine transform

Fs {e^-ax } = w/(a² + w²)

here a=pi

Fs {e^-pi*x } = w/(pi² + w²)

d]   Fs { x^-n } = cos(0.5*pi*n)* T(n-1) *w^n-1

     here n= 1/2

Fs { x^-0.5} = cos(0.5*pi*1/2)* T(1/2-1) *w^1/2-1 =  cos(pi/4)* T(1/2) *w^-1/2 = 1/sqrt(2) * sqrt (pi) w^-1/2

     = sqrt( pi / (2*W) )