Fourier transform of a translated variable

Hello,

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:

By definition,

Translating the variable to , we get,

Thus,

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.

Thanks,