We define $x(t)\xrightarrow{\mathscr{F}}X(j\omega)$
Assume I have that : $$ X(j\omega)=\begin{cases}e^{-2\omega}&\text{if $|\omega|<2\pi$}\\0&\text{if otherwise}\end{cases} $$ Say, I want to make this change substitution $X(j(\omega-\Omega))$. Would this imply : $$ X(j(\omega-\Omega))=\begin{cases}e^{-2(\omega-j\Omega)}&\text{if $-j(2\pi+\omega)<\Omega<-j(\omega-2\pi)$}\\0&\text{if otherwise}\end{cases}\;\;??? $$ I am asking this question since I wish to compute the modulation : $$ X_{1}*X_{2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}X_{1}(j\Omega)X_{2}(j(\omega-\Omega))\;\text{d}\Omega $$ and as you can see here, there is a change of variable for $X_{2}$