Prove the following properties of characteristic functions:

(a) Every characteristic function has value unity at the origin.

(b) The second-order characteristic function \(\mathbf{M}_{U V}\left(\omega_{U}, 0\right)\) is equal to the characteristic function \(\mathbf{M}_{U}(\omega)\) of the random variable \(U\) alone.

(c) For two independent random variables \(U\) and \(V\),

\[ \mathbf{M}_{U V}\left(\omega_{U}, \omega_{V}\right)=\mathbf{M}_{U}\left(\omega_{U}\right) \mathbf{M}_{V}\left(\omega_{V}\right) \]