Editing Stable distribution (section)

== Other analytic cases ==

A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by <math>f(x;\alpha,\beta,c,\mu)</math>, then:

* The [[Cauchy Distribution]] is given by <math>f(x;1,0,1,0).</math>
* The [[Lévy distribution]] is given by <math>f(x;\tfrac{1}{2},1,1,0).</math>
* The [[Normal distribution]] is given by <math>f(x;2,0,1,0).</math>
* Let <math>S_{\mu,\nu}(z)</math> be a [[Lommel function]], then:<ref name="Garoni2002">{{cite journal |last1=Garoni |first1=T. M. |last2=Frankel |first2=N. E. |date=2002 |title=Lévy flights: Exact results and asymptotics beyond all orders |journal=Journal of Mathematical Physics |volume=43 |issue=5 |pages=2670–2689 |doi= 10.1063/1.1467095|bibcode=2002JMP....43.2670G}}</ref> <math display="block"> f \left (x;\tfrac{1}{3},0,1,0\right ) = \Re\left ( \frac{2e^{- \frac{i \pi}{4}}}{3 \sqrt{3} \pi}  \frac{1}{\sqrt{x^3}} S_{0,\frac{1}{3}} \left (\frac{2e^{\frac{i \pi}{4}}}{3 \sqrt{3}}  \frac{1}{\sqrt{x}} \right) \right )</math>
* Let <math>S(x)</math> and <math>C(x)</math> denote the [[Fresnel integral]]s, then:<ref name="Hopcraft1999">{{cite journal |last1=Hopcraft |first1=K. I. |last2=Jakeman |first2=E.|last3=Tanner|first3=R. M. J. |date=1999 |title=Lévy random walks with fluctuating step number and multiscale behavior |journal=Physical Review E |volume=60 |issue=5 |pages=5327–5343 |doi= 10.1103/physreve.60.5327|pmid=11970402 |bibcode=1999PhRvE..60.5327H}}</ref> <math display="block">f\left (x;\tfrac{1}{2},0,1,0\right ) = \frac{1}{{\sqrt{2\pi|x|^3}}}\left (\sin\left(\tfrac{1}{4|x|}\right) \left [\frac{1}{2} - S\left (\tfrac{1}{\sqrt{2\pi|x|}}\right )\right ]+\cos\left(\tfrac{1}{4|x|} \right) \left [\frac{1}{2}-C\left (\tfrac{1}{\sqrt{2\pi|x|}}\right )\right ]\right )</math>
* Let <math>K_v(x)</math> be the [[modified Bessel function]] of the second kind, then:<ref name="Hopcraft1999"/> <math display="block">f\left (x;\tfrac{1}{3},1,1,0\right ) = \frac{1}{\pi} \frac{2\sqrt{2}}{3^{\frac{7}{4}}} \frac{1}{\sqrt{x^3}} K_{\frac{1}{3}}\left (\frac{4\sqrt{2}}{3^{\frac{9}{4}}} \frac{1}{\sqrt{x}} \right )</math>
* Let <math>{}_mF_n</math> denote the [[hypergeometric function]]s, then:<ref name="Garoni2002"/> <math display="block">\begin{align}
 f\left (x;\tfrac{4}{3},0,1,0\right ) &= \frac{3^{\frac{5}{4}}}{4 \sqrt{2 \pi}} \frac{\Gamma \left (\tfrac{7}{12} \right ) \Gamma \left (\tfrac{11}{12} \right )}{\Gamma\left (\tfrac{6}{12} \right ) \Gamma \left (\tfrac{8}{12} \right )} {}_2F_2 \left ( \tfrac{7}{12}, \tfrac{11}{12}; \tfrac{6}{12}, \tfrac{8}{12}; \tfrac{3^3 x^4}{4^4} \right ) - \frac{3^{\frac{11}{4}}x^3}{4^3 \sqrt{2 \pi}} \frac{\Gamma \left (\tfrac{13}{12} \right ) \Gamma \left (\tfrac{17}{12} \right )}{\Gamma \left (\tfrac{18}{12} \right ) \Gamma \left (\tfrac{15}{12} \right )} {}_2F_2 \left ( \tfrac{13}{12}, \tfrac{17}{12}; \tfrac{18}{12}, \tfrac{15}{12}; \tfrac{3^3 x^4}{4^4} \right ) \\[6pt]
f\left (x;\tfrac{3}{2},0,1,0\right ) &= \frac{\Gamma \left(\tfrac{5}{3} \right)}{\pi} {}_2F_3 \left ( \tfrac{5}{12}, \tfrac{11}{12}; \tfrac{1}{3}, \tfrac{1}{2}, \tfrac{5}{6}; - \tfrac{2^2 x^6}{3^6} \right )
- \frac{x^2}{3 \pi} {}_3F_4 \left ( \tfrac{3}{4}, 1, \tfrac{5}{4}; \tfrac{2}{3}, \tfrac{5}{6}, \tfrac{7}{6}, \tfrac{4}{3}; - \tfrac{2^2 x^6}{3^6} \right ) + \frac{7 x^4\Gamma  \left(\tfrac{4}{3} \right)}{3^4 \pi ^ 2} {}_2F_3 \left ( \tfrac{13}{12}, \tfrac{19}{12}; \tfrac{7}{6}, \tfrac{3}{2}, \tfrac{5}{3}; -\tfrac{2^2 x^6}{3^6} \right)
\end{align}</math>  with the latter being the [[Holtsmark distribution]].
* Let <math>W_{k,\mu}(z)</math> be a [[Whittaker function]], then:<ref name="Zolotarev1999">{{cite journal |last1=Uchaikin |first1=V. V. |last2=Zolotarev |first2=V. M. |date=1999 |title=Chance And Stability – Stable Distributions And Their Applications |journal=VSP }}</ref><ref>{{cite journal |last=Zlotarev |first=V. M. |date=1961 |title=Expression of the density of a stable distribution with exponent alpha greater than one by means of a frequency with exponent 1/alpha |journal=Selected Translations in Mathematical Statistics and Probability (Translated from the Russian Article: Dokl. Akad. Nauk SSSR. 98, 735–738 (1954)) | volume=1 |pages=163–167 }}</ref><ref>{{cite journal |last1=Zaliapin |first1=I. V. |last2=Kagan |first2=Y. Y. |last3=Schoenberg | first3=F. P. |date=2005 |title=Approximating the Distribution of Pareto Sums |url= http://www.escholarship.org/uc/item/8940b4k8 | journal=Pure and Applied Geophysics |volume=162 |issue=6 |pages=1187–1228 |doi= 10.1007/s00024-004-2666-3 | bibcode=2005PApGe.162.1187Z |s2cid=18754585 }}</ref> <math display="block">\begin{align}
f\left (x;\tfrac{2}{3},0,1,0\right ) &= \frac{\sqrt{3}}{6\sqrt{\pi}|x|} \exp\left (\tfrac{2}{27}x^{-2}\right ) W_{-\frac{1}{2},\frac{1}{6}}\left (\tfrac{4}{27}x^{-2}\right ) \\[8pt]
f\left (x;\tfrac{2}{3},1,1,0\right ) &= \frac{\sqrt{3}}{\sqrt{\pi}|x|} \exp\left (-\tfrac{16}{27}x^{-2}\right ) W_{\frac{1}{2},\frac{1}{6}} \left (\tfrac{32}{27}x^{-2}\right ) \\[8pt]
f\left (x;\tfrac{3}{2},1,1,0\right ) &= \begin{cases} \frac{\sqrt{3}}{\sqrt{\pi}|x|} \exp\left (\frac{1}{27}x^3\right ) W_{\frac{1}{2},\frac{1}{6}}\left (- \frac{2}{27}x^3\right ) & x<0\\ {} \\ \frac{\sqrt{3}}{6\sqrt{\pi}|x|} \exp\left (\frac{1}{27}x^3\right ) W_{-\frac{1}{2},\frac{1}{6}}\left (\frac{2}{27}x^3\right ) & x \geq 0 \end{cases}
\end{align}</math>