How to find central moment from characteristic function

Question

Suppose I have $\int\limits_{0}^{1} W^2(t) dt =: W_1$, with characteristic function $$ E \left[ e^{izW_1} \right] = \left( \cos \sqrt{2iz} \right)^{-\frac{1}{2}} =: F(z) .$$ I can calculate first, second and other moments because of theorem $$\varphi_X^{(k)}(0) = i^kE[X^k].$$ However I can't find theorem to calculate central moment (to find variance). Is there any possibility to calculate second central moment from characteristic function?

Answer 1

$$E[e^{izW_1}] = \cos(\sqrt{2iz})^{-1/2} = \cosh(\sqrt{2z})^{-1/2} = \Big( \sum_{n=0}^{+\infty} \frac{(2z)^n}{(2n)!}\Big)^{-1/2}.$$ This shows that the characteristic function is analytic and enables us to compute the first derivatives at $0$. Then, you may use $\mathrm{Var}(W_1) = E[W_1^2] - E[W_1]^2$.