The properties and evaluation of an integral with power singularities
The paper is devoted to the computation of integrals of \(f_ \alpha (t):= t^{\alpha-1} (t-1)^{-\alpha} (1-kt)^{\alpha-1}\). The authors present numerical methods for the approximate evaluation of \(\int^{1/k}_ 1 f_ \alpha (u)du\) and \(A_ \alpha (t):=\int^ t_ 1f_ \alpha (u)du\) and give precise estimations of the remainder. The behavior of \(A_ \alpha (t)\) is studied for limit values of \(k\) and \(\alpha\) \((k \to 0,\;k \to 1,\;\alpha \to 0)\).
scientific article; zbMATH DE number 486820