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The Fox-Wright function is a further extension of the generalizedhypergeometric function obtained by introducing arbitrary positive scalingfactors into the arguments of the gamma functions in the summand. Itsimportance comes mostly from its role in fractional calculus although otherinteresting applications also exist. If the sums of the scaling factors in thetop and bottom parameters are equal, the series defining the Fox-Wrightfunction has a finite non-zero radius of convergence. It was demonstrated byBraaksma in 1964 that the Fox-Wright function can then be extended to aholomorphic function in the complex plane cut along a ray from the positivepoint on the boundary of the disk of convergence to the point at infinity. Inthis paper we study the behavior of the Fox-Wright function in the neighborhoodof this positive singular point. Under certain restrictions we give aconvergent expansion with recursively computed coefficients completelycharacterizing this behavior. We further compute the jump and the average valueof the Fox-Wright function on the banks of the branch cut.

The Fox-Wright function near the singularity and branch cut