Suppose I have:

$\displaystyle f(x)=(1-|x|)1_{(-1,1)}(x)$

I want to show that its characteristic function is:

$\displaystyle \varphi (u)= 2(1-cosu)/u^2$

I let

$\displaystyle E(e^{iux})=\int_{-1}^1e^{iux}(1-|x|).dx$

$\displaystyle =\int_{-1}^0e^{iux}(1+x).dx+\int_{0}^1e^{iux}(1-x).dx$

...

$\displaystyle =\frac{1}{(iu)^2}(e^{-iu}+e^{iu})$

$\displaystyle =2(\cos u)/u^2$

May I know what went wrong?