In this paper we discuss analytical and numerical properties of the function $V_{\nu,\mu}(\alpha,\beta,z)=\int_0^\infty e^{-zt}(t+\alpha)^\nu(t+\beta)^\mu\,dt$, with $\alpha,\beta,\Re z>0$, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer $U-$function. The function $V_{\nu,\mu}(\alpha,\beta,z)$ is used for certain values of the parameters as an approximant in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.

Additional Metadata
Keywords complementary error function, singular perturbation problem, incomplete gamma function, confluent hypergeometric function, asymptotic expansions, recurrence relations
MSC Computation of special functions, construction of tables (msc 65D20)
Publisher Elsevier
Journal Applied Mathematics and Computation
Note doi:10.1016/j.amc.2010.05.025
Deaño, A, & Temme, N.M. (2010). Analytical and Numerical Aspects of a Generalization of the Complementary Error Function. Applied Mathematics and Computation.