I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to gaussian after this inverse fourier transform. ) I used this inverse fourier formula
$$ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$$
I used euler formula for this integral $$ e^{i \ x} = cos(x)+isin(x) $$
so i get $$ e^{i \ s\ x} = cos(sx) + sin(sx) $$
then i put $$cos(sx) + sin(sx) $$ in place of $$e^{ i \ s\ x}$$ , so that i get two integrals.
$$ f(x) =\int^{\infty}_{-\infty} e^{-2\pi^2/s^2} cos(sx)dx +i \int^{\infty}_{-\infty}e^{-2\pi^2/s^2} sin(sx)ds $$
I use then $$ uv - \int^{}_{}vdu$$
each time i solve these integrals i get 0 as result. I cant see what i am doing wrong?