Combining risks III: characteristic functions of standard probability distributions

What is this page?

These are my notes to myself on the characteristic functions of some standard probability distributions. I'm just leaving it here in case other people find it useful. It was generated by AI.

Characteristic functions of continuous distributions

\(\varphi_X(t) = \mathbb{E}\!\left[e^{itX}\right]\), with all distributions in their standard parameterizations.

\(\text{Distribution}\)	Characteristic function \(\varphi(t)\)	What the function is
\(\text{Arcsine on } [0,1]\)	\(e^{it/2}\,J_0\!\left(\tfrac{t}{2}\right)\)	Phase factor \(e^{it/2}\) times the Bessel function of the first kind \(J_0\)
\(\text{Beta}(\alpha,\beta)\)	\({}_1F_1(\alpha;\ \alpha+\beta;\ it)\)	Kummer's confluent hypergeometric function \({}_1F_1(a;b;z)\)
\(\text{Cauchy}(x_0,\gamma)\)	\(e^{\,i x_0 t - \gamma|t|}\)	Linear phase times \(e^{-\gamma|t|}\), a Laplace-shaped function of \(t\) (the Fourier dual of the Cauchy density)
\(\text{Chi}(k)\)	\(\begin{aligned}&{}_1F_1\!\left(\tfrac{k}{2};\tfrac{1}{2};-\tfrac{t^2}{2}\right)\\&+\,it\sqrt{2}\,\tfrac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}\\&\quad\times{}_1F_1\!\left(\tfrac{k+1}{2};\tfrac{3}{2};-\tfrac{t^2}{2}\right)\end{aligned}\)	Sum of two Kummer functions \({}_1F_1\) (an even real part and an odd imaginary part in \(t\))
\(\text{Chi-squared}\ (\chi^2_k)\)	\((1-2it)^{-k/2}\)	Complex power function \((1-2it)^{-k/2}\) (Gamma-type)
\(\text{Erlang}(k,\lambda)\)	\(\left(1-\dfrac{it}{\lambda}\right)^{-k}\)	Rational function \(\left(1-it/\lambda\right)^{-k}\), a negative integer power of a linear term
\(\text{Exponential}(\lambda)\)	\(\left(1-\dfrac{it}{\lambda}\right)^{-1}\)	Lorentzian-type rational function with a single pole at \(t=-i\lambda\)
\(\text{Gamma}(k,\lambda)\text{ (shape, rate)}\)	\(\left(1-\dfrac{it}{\lambda}\right)^{-k}\)	Complex power function \(\left(1-it/\lambda\right)^{-k}\) (Gamma-type)
\(\text{Gumbel}(\mu,\beta)\)	\(\Gamma(1-i\beta t)\,e^{i\mu t}\)	Gamma function \(\Gamma(z)\) of a complex argument, times a linear phase
\(\text{Half-normal}(\sigma)\)	\(\begin{aligned}&e^{-\sigma^2 t^2/2}\\&\times\left[1+i\operatorname{erfi}\!\left(\tfrac{\sigma t}{\sqrt{2}}\right)\right]\end{aligned}\)	Gaussian \(e^{-\sigma^2t^2/2}\) multiplied by \(1+i\operatorname{erfi}(\cdot)\), where \(\operatorname{erfi}\) is the imaginary error function
\(\text{Hyperbolic secant}\)	\(\operatorname{sech}(t)\)	Hyperbolic secant \(\operatorname{sech}(t)\); the density is self-dual under the Fourier transform
\(\text{Inverse-gamma}(\alpha,\beta)\)	\(\dfrac{2\,(-i\beta t)^{\alpha/2}}{\Gamma(\alpha)}\,K_{\alpha}\!\left(\sqrt{-4i\beta t}\right)\)	Modified Bessel function of the second kind \(K_\alpha\) times a complex power
\(\text{Inverse Gaussian}(\mu,\lambda)\)	\(\exp\!\left[\tfrac{\lambda}{\mu}\left(1-\sqrt{1-\tfrac{2\mu^2 it}{\lambda}}\right)\right]\)	Exponential of a square root, \(\exp\!\big(c\,(1-\sqrt{1-z})\big)\)
\(\text{Irwin–Hall}(n)\)	\(\left(\dfrac{e^{it}-1}{it}\right)^{n}\)	The \(n\)th power of a phase-shifted \(\operatorname{sinc}\) function
\(\text{Laplace}(\mu,b)\)	\(\dfrac{e^{i\mu t}}{1+b^2t^2}\)	Lorentzian \(\dfrac{1}{1+b^2t^2}\) times a linear phase
\(\text{Lévy}(\mu,c)\)	\(\exp\!\left(i\mu t - \sqrt{-2ict}\right)\)	Exponential of a square root (stable law with \(\alpha=\tfrac12\))
\(\text{Logistic}(\mu,s)\)	\(e^{i\mu t}\,\dfrac{\pi s t}{\sinh(\pi s t)}\)	\(x/\sinh x\) (hyperbolic cosecant type) times a linear phase
\(\text{Noncentral chi-squared}(k,\lambda)\)	\(\dfrac{\exp\!\left(\dfrac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}\)	Exponential of a rational function times a complex power (a Poisson mixture of \(\chi^2\) laws)
\(\text{Normal}(\mu,\sigma^2)\)	\(\exp\!\left(i\mu t - \tfrac{1}{2}\sigma^2 t^2\right)\)	Gaussian function \(e^{-\sigma^2t^2/2}\) times a linear phase
\(\text{Raised cosine}(\mu,s)\)	\(e^{i\mu t}\,\dfrac{\pi^2\sin(st)}{st\,(\pi^2 - s^2t^2)}\)	\(\operatorname{sinc}\)-type function with an extra factor \((\pi^2-s^2t^2)\) in the denominator, times a linear phase
\(\text{Rayleigh}(\sigma)\)	\(\begin{aligned}1-{}&\sigma t\,e^{-\sigma^2t^2/2}\sqrt{\tfrac{\pi}{2}}\\&\times\left[\operatorname{erfi}\!\left(\tfrac{\sigma t}{\sqrt{2}}\right)-i\right]\end{aligned}\)	Gaussian \(e^{-\sigma^2t^2/2}\) combined with the imaginary error function \(\operatorname{erfi}\)
\(\text{Semicircle (Wigner), radius } R\)	\(\dfrac{2\,J_1(Rt)}{Rt}\)	Bessel function of the first kind over its argument, \(2J_1(x)/x\) (jinc-type)
\(\text{Stable}(\alpha,\beta,c,\mu)\)	\(\exp\!\left[i\mu t - |ct|^{\alpha}\left(1 - i\beta\,\operatorname{sgn}(t)\,\Phi\right)\right]\),  \(\Phi=\tan\tfrac{\pi\alpha}{2}\) if \(\alpha\ne1\), \(\Phi=-\tfrac{2}{\pi}\log|t|\) if \(\alpha=1\)	Exponential of a power of \(|t|\), \(\exp\!\left(-|ct|^{\alpha}\cdots\right)\)
\(\text{Student's } t\,(\nu)\)	\(\dfrac{\left(\sqrt{\nu}\,|t|\right)^{\nu/2}K_{\nu/2}\!\left(\sqrt{\nu}\,|t|\right)}{\Gamma(\nu/2)\,2^{\nu/2-1}}\)	Modified Bessel function of the second kind \(K_{\nu/2}\) times a power of \(|t|\)
\(\text{Triangular}(a,b,c)\)	\(\dfrac{\begin{aligned}-2\big[&(b-c)e^{iat}-(b-a)e^{ict}\\&+(c-a)e^{ibt}\big]\end{aligned}}{(b-a)(c-a)(b-c)\,t^2}\)	Sum of complex exponentials divided by \(t^2\) (squared-\(\operatorname{sinc}\)-type)
\(\text{Truncated normal}(\mu,\sigma^2,a,b)\)	\(\begin{aligned}&e^{i\mu t-\sigma^2t^2/2}\\&\times\dfrac{\Phi(\beta-i\sigma t)-\Phi(\alpha-i\sigma t)}{\Phi(\beta)-\Phi(\alpha)}\end{aligned}\), where \(\alpha=\tfrac{a-\mu}{\sigma}\), \(\beta=\tfrac{b-\mu}{\sigma}\), and \(\Phi\) is the standard normal CDF (extended to complex arguments)	Gaussian function \(e^{-\sigma^2t^2/2}\) times a ratio of differences of \(\Phi\) at complex arguments (equivalently, of error functions \(\operatorname{erf}\)), times a linear phase
\(\text{Uniform}(a,b)\)	\(\dfrac{e^{itb}-e^{ita}}{it\,(b-a)}\)	\(\operatorname{sinc}\) function (difference of exponentials over \(t\)) times a linear phase

Notation: \(J_\nu\) is the Bessel function of the first kind, \(K_\nu\) the modified Bessel function of the second kind, \({}_1F_1\) Kummer's confluent hypergeometric function, and \(\operatorname{erfi}(x) = -i\operatorname{erf}(ix)\) the imaginary error function. Complex powers and square roots use the principal branch. Values at \(t=0\) are understood as limits (\(\varphi(0)=1\)). Distributions without a closed form (e.g. log-normal, Weibull, Pareto) are omitted.