How to solve the Fast Fourier transform?

Inverse Fourier Transform by Residue Theorem and Contour Integration?

• Q) Use the Residue theorem and contour integration to show that the inverse Fourier transform of the function F(w) = (1/Sqrt[2*pi]) * (1+jw) / (1+w^2) is given by the function f(x)= e^x for x<0; 1/2 for x=0; 0 for otherwise ----------------------- My current strategy is to integrate about a closed contour around the pole at w=-j, which would then equal 2*pi*j times the residue at that point (e^x). This would then equal the integral along the real axis between -R and R, plus the integral along the open path given by the semi circle under the real axis which encircles the pole at -j. I would then rearrange to solve for the desired integral. My question is, am I going about this the right way? I've attempted to follow the method described above, but it leads to what seems to be a dead end (I get a horrific integral with exponentials of exponentials)... Any help would be much appreciated!

• Answer:

  i've been trying to answer same question so if you sort it out pliz give me a shout.