Find the characteristic function for the following probability density functions: (a) (ho(x)=frac{a}{2} e^{-a|x|}); (b) (p(x)=frac{a}{pileft(a^{2}+x^{2} ight)}); (c) (p(x)=left{begin{array}{c}0 text { when }|x| geqslant a

Find the characteristic function for the following probability density functions:

(a) \(ho(x)=\frac{a}{2} e^{-a|x|}\);

(b) \(p(x)=\frac{a}{\pi\left(a^{2}+x^{2}\right)}\);

(c) \(p(x)=\left\{\begin{array}{c}0 \text { when }|x| \geqslant a \\ \frac{a-|x|}{a^{2}} \text { when }|x| \leqslant a ;\end{array}\right.\)

(d) \(p(x)=\frac{2 \sin ^{2} \frac{a x}{2}}{\pi a x^{2}}\)

The attentive reader will have noted that Examples (a) and (b) and also (c) and (d) are, so to speak, inverse.