Question

Find the fourier transform of f(x)=1/[x(x^2+a^2)]​

Answer 1

Answer:

Consider the function f(x)=e−a|x|. Then

f^(ω)=∫∞−∞e−a|x|e−iωxdx=∫0−∞eaxe−iωxdx+∫∞0e−axe−iωxdx==[e(a−iω)xa−iω]0−∞−[e−(a+iω)xa+iω]∞0=1a−iω+1a+iω=2aa2+ω2

Now, by the inversion forumla, we have

e−a|x|=12π∫∞−∞2aa2+ω2eiωxdω

Changing the sign on x and multiplying by πa, we finally get

πae−a|−x|=πae−a|x|=∫∞−∞e−iωxa2+ω2dω

Thus,

b^(ω)=πae−a|ω|