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The Wikipedia article on the distribution also doesn't seem to specify any formulae on right hand side. Also wolframalpha says there is no closed form solution.

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    $\begingroup$ I haven't worked it out, but perhaps there is something useful at apesimulator.org/help/models/rain/… , since it appears that the author is trying to use the inverse distribution method to generate random variates? Or alternatively, use a numerical nonlinear equation solver to numerically determine the inverse for a given argument. $\endgroup$ Jul 18, 2016 at 16:33
  • $\begingroup$ I used wolfram alpha (also updated in the question), it couldn't find any closed form solution. $\endgroup$ Jul 19, 2016 at 4:50

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The inverse of the CDF is the quantile function. There is no closed form solution but it is already implemented in the R package sn. The way the package calculates the quantile of interest is by solving the non-linear equation $F(q) = p$ numerically.

https://cran.r-project.org/web/packages/sn/index.html

Description

Density function, distribution function, quantiles and random number generation for the skew-normal (SN) and the extended skew-normal (ESN) distribution.

Usage

dsn(x, xi=0, omega=1, alpha=0, tau=0, dp=NULL, log=FALSE) psn(x, xi=0, omega=1, alpha=0, tau=0, dp=NULL, engine, ...) qsn(p, xi=0, omega=1, alpha=0, tau=0, dp=NULL, tol=1e-8, solver="NR", ...) rsn(n=1, xi=0, omega=1, alpha=0, tau=0, dp=NULL)

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