Fourier transform of a translated variable
I would like to derive the Fourier transform of a translated variable. I read in some references (including wikipedia) that,
where is the translation operator translating the scalar or vector field by , is the wave number, is the imaginary number and denotes the fourier transform.
I have tried deriving it myself but have some doubts. Following is the derivation:
Translating the variable to , we get,
My questions are:
- Is my derivation correct? If not, then how to derive it?
- If it is correct, then while translating by , shouldn't the exponential term also be translated, which means that the exponential term will be ?
Please tell me what am I doing wrong or where I am getting confused.