Hello,

I would like to derive the Fourier transform of a translated variable. I read in some references (including wikipedia) that,

\widehat{T_{\delta_{x}}v(k)} = e^{-\mathrm{i}k\delta_{x}}\widehat{v(k)},
where T_{\delta_{x}} is the translation operator translating the scalar or vector field v(x) by \delta x, k is the wave number, \mathrm{i} is the imaginary number and ^{\widehat{}} denotes the fourier transform.

I have tried deriving it myself but have some doubts. Following is the derivation:
By definition, \widehat{v(k)} = \int_{-\infty}^{\infty} v(x) e^{\mathrm{i}kx} dx
=  \int_{-\infty}^{\infty} v(x) e^{\mathrm{i}kx} e^{\mathrm{i}k\delta x} e^{-\mathrm{i}k\delta x} dx
=  e^{-\mathrm{i}k\delta x}  \int_{-\infty}^{\infty} v(x) e^{\mathrm{i}k(x+\delta x)} dx
Translating the variable x to x+\delta x, we get,
=  e^{-\mathrm{i}k\delta x}  \int_{-\infty}^{\infty} v(x+\delta x) e^{\mathrm{i}k(x+\delta x)} d(x+\delta x)
Thus,
\widehat{T_{\delta_{x}}v(k)} = e^{-\mathrm{i}k\delta_{x}}\widehat{v(k)}

My questions are:
  1. Is my derivation correct? If not, then how to derive it?
  2. If it is correct, then while translating x by x+\delta x, shouldn't the exponential term also be translated, which means that the exponential term will be e^{\mathrm{i}k(x+2\delta x)} ?


Please tell me what am I doing wrong or where I am getting confused.

Thanks,