The parameterized-Euler-constant function γα(z)

We define parameterized-Euler-constant function with parameter α>−1, defined on the closed unit disk. The function γα(z) is an extension of the function z↦γ(z):=γ1(z) studied recently and having the property that γ1(1)=C, C being the Euler–Mascheroni constant. Integrals for γα(z) provide its analytic continuation to C∖[1,+∞).