Editing Stable distribution (section)

==Series representation==
The stable distribution can be restated as the real part of a simpler integral:<ref name=":3">{{Cite journal|title = Theory of the pressure broadening and shift of spectral lines|journal = Advances in Physics|date = 1981|issn = 0001-8732|pages = 367–474 | volume = 30|issue = 3|doi = 10.1080/00018738100101467|first = G.|last = Peach|bibcode = 1981AdPhy..30..367P}}</ref>
<math display="block">f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \int_0^\infty e^{it(x-\mu)}e^{-(ct)^\alpha(1-i\beta\Phi)}\,dt\right].</math>

Expressing the second exponential as a [[Taylor series]], this leads to:
<math display="block">f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \int_0^\infty e^{it(x-\mu)}\sum_{n=0}^\infty\frac{(-qt^\alpha)^n}{n!}\,dt\right]</math>
where <math>q=c^\alpha(1-i\beta\Phi)</math>. Reversing the order of integration and summation, and carrying out the integration yields:
<math display="block">f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \sum_{n=1}^\infty\frac{(-q)^n}{n!}\left(\frac{i}{x-\mu}\right)^{\alpha n+1}\Gamma(\alpha n+1)\right]</math>
which will be valid for ''x''&nbsp;≠&nbsp;''μ'' and will converge for appropriate values of the parameters. (Note that the ''n''&nbsp;=&nbsp;0 term which yields a [[Dirac delta function|delta function]] in ''x''&nbsp;−&nbsp;''μ'' has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of ''x''&nbsp;−&nbsp;''μ'' which is generally less useful.

For one-sided stable distribution, the above series expansion needs to be modified, since <math>q=\exp(-i\alpha\pi/2)</math> and <math>q i^{\alpha}=1</math>. There is no real part to sum. Instead, the integral of the characteristic function should be carried out on the negative axis, which yields:<ref>{{Cite journal|last=Pollard|first=Howard|date=1946|title=<nowiki>Representation of e^{-x^\lambda} As a Laplace Integral</nowiki>|url=https://projecteuclid.org/euclid.bams/1183509728|journal=Bull. Amer. Math. Soc. |volume=52|pages=908|doi=10.1090/S0002-9904-1946-08672-3|via=|doi-access=free}}</ref><ref name="PhysRevLett 1007"/>
<math display="block">\begin{align}
L_\alpha(x) & =
\frac{1}{\pi}\Re\left[ \sum_{n=1}^\infty\frac{(-q)^n}{n!}\left(\frac{-i}{x}\right)^{\alpha n+1}\Gamma(\alpha n+1)\right]
\\ & =
\frac{1}{\pi}\sum_{n=1}^\infty\frac{-\sin(n(\alpha+1)\pi)}{n!}\left(\frac{1}{x}\right)^{\alpha n+1}\Gamma(\alpha n+1)
\end{align} </math>