If a < 0 then the limits in the integral will be reversed since , i.e. you'll have instead of
Switching the limits back will also switch the sign of the integral.
I am having trouble with the proof of the time shifting property in the fourier transform. The property states that if x(t) has a fourier transform X(jw) then, x(at) has a transform (abs(a^-1)X(jw/a). My problem is with the abs!! why does it have a abs there? The proof of the property is in a file stored on dropbox( http://dl.dropbox.com/u/27392043/ooooo.jpg ) because i cant upload it here dunno why. If you look at the picture what I dont understand is the - signal where a<0 why is it there if when T=at dT/a=dt never dT/-a=dt!!! please help me.
Thanks.