I'm trying to derive the characteristic function for the Laplace distribution with density $$\frac{1}{2}\exp\{-|x|\}$$
My attempt:
$$\frac{1}{2}\int_{\Omega}e^{itx-|x|}\mathrm{d}x$$
$$=\frac{1}{2}\int_0^\infty e^{(-it+1)-x}\mathrm{d}x+\frac{1}{2}\int_{-\infty}^0e^{(it+1)x}\mathrm{d}x$$
$$=\frac{1}{2}e^{(-it+1)}\int_0^\infty e^{-x}\mathrm{d}x+\frac{1}{2}e^{(it+1)}\int_{-\infty}^0e^{x}\mathrm{d}x$$
$$\frac{1}{2}e^{-it+1}+\frac{1}{2}e^{(it+1)}$$
$$\frac{1}{2}e(e^{-it}+e^{it})$$
But this doesn't look like the characteristic function on Wikipedia.
Best Answer
In the third line you wrote substantially that $$ e^{(it+1)x}=e^{(it+1)x}e^{x} $$ which is extremely wrong. In the second line you also wrote $e^{(-i t +1)-x} $ which is inconsistent with the first line.