For any random variable $X$ with probability measure $\mu$ and distribution function $F$, the characteristic function (ch.f.) is a function $f$ on $\mathbb{R}$ defined as $$f(t)=\mathscr{E}\left(e^{itX}\right)=\int_{-\infty}^\infty e^{itx}\,dF(x)\mbox{ for all }t\in\mathbb{R}.$$There are some simple properties of ch.f.:
(1) For all $t\in\mathbb{R}$, $$|f(t)|\leq1=f(0);\quad f(-t)=\overline{f(t)},$$where $\overline{z}$ denotes the conjugate complex of $z$.
(2) $f$ is uniformly continuous in $\mathbb{R}$.
(3) Write $f_X(t)$ for the ch.f. of $X$, then for any real numbers $a$ and $b$, we have $$f_{aX+b}(t)=e^{itb}f_X(at),$$ $$f_{-X}(t)=\overline{f_X(t)}.$$
(4) If $\{f_n,n\geq1\}$ are ch.f.'s, $\lambda_n\geq0$ and $\sum_{n=1}^\infty\lambda_n=1$, then $$\sum_{n=1}^\infty\lambda_nf_n$$is a ch.f. (a convex combination of ch.f.'s is a ch.f.). Further, if $\{\mu_n,n\geq1\}$ are the corresponding p.m.'s, then $\sum_{n=1}^\infty\lambda_n\mu_n$ is a p.m. whose ch.f. is $\sum_{n=1}^\infty\lambda_nf_n$.
(5) If $\{f_j,1\leq j\leq n\}$ are ch.f.'s, then $$\prod_{j=1}^nf_j\mbox{ is a ch.f.}$$
Here are some well-known ch.f.'s:
First, define $\delta_a$ be an indicator function that $\delta_a(x)=1$, if $x=a$, $\delta_a(x)=0$, otherwise.
(i) Point mass at $a$: $$\mbox{d.f. } \delta_a;\quad\mbox{ch.f. }e^{iat}.$$
(ii) Symmetric Bernoulian distribution with mass $\frac{1}{2}$ each at $+1$ and $-1$: $$\mbox{d.f. } \frac{1}{2}(\delta_1+\delta_{-1});\quad\mbox{ch.f. }\cos{t}.$$
(iii) Bernoullian distribution with "success probability" $p$ and $q=1-p$: $$\mbox{d.f. } q\delta_0+p\delta_1);\quad\mbox{ch.f. }q+pe^{it}=1+p\left(e^{it}-1\right).$$
(iv) Binomial distribution for $n$ trials with success probability $p$: $$\mbox{d.f. } \sum_{k=0}^n{n\choose k}p^kq^{n-k}\delta_k;\quad\mbox{ch.f. }\left(q+pe^{it}\right)^n.$$
(v) Geometric distribution with success probability $p$: $$\mbox{d.f. } \sum_{n=0}^\infty q^np\delta_n;\quad\mbox{ch.f. }p\left(1-qe^{it}\right)^{-1}.$$
(vi) Possion distribution with mean $\lambda$: $$\mbox{d.f. } \sum_{n=0}^\infty e^{-\lambda}\frac{\lambda^n}{n!}\delta_n;\quad\mbox{ch.f. }\exp{\left\{\lambda\left(e^{it}-1\right)\right\}}.$$
(vii) Exponential distribution with mean $\lambda^{-1}$: $$\mbox{p.d. }\lambda e^{-\lambda x}\mbox{ in }[0,\infty);\quad\mbox{ch.f. }\left(1-\lambda^{-1}it\right)^{-1}.$$
(viii) Uniform distribution in $[-a,+a]$: $$\mbox{p.d. }\frac{1}{2a}\mbox{ in }[-a,+a];\quad\mbox{ch.f. }\frac{\sin{at}}{at},\,=1\mbox{ for }t=0.$$
(ix) Triangular distribution in $[-a,+a]$: $$\mbox{p.d. }\frac{a-|x|}{a^2}\mbox{ in }[-a,+a];\quad\mbox{ch.f. }\frac{2(1-\cos{at})}{a^2t^2}=\left(\frac{\sin{(at/2)}}{(at/2)}\right)^2.$$
(x) Reciprocal of triangular distribution: $$\mbox{p.d. }\frac{1-\cos{ax}}{\pi ax^2}\mbox{ in }\mathbb{R};\quad\mbox{ch.f. }\left(1-\frac{|t|}{a}\right)\vee0.$$
(xi) Normal distribution $N(m,\sigma^2)$: $$\mbox{p.d. }\frac{1}{\sqrt{2\pi}\sigma}\exp{\left\{-\frac{(x-m)^2}{2\sigma^2}\right\}}\mbox{ in }\mathbb{R};\quad\mbox{ch.f. }\exp{\left\{imt-\frac{\sigma^2t^2}{2}\right\}}.$$
(xii) Cauchy distribution with parameter $a>0$: $$\mbox{p.d. }\frac{a}{\pi(a^2+x^2)}\mbox{ in }\mathbb{R};\quad\mbox{ch.f. }e^{-a|t|}.$$
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