Nu function

In mathematics, the nu function is a generalization of the reciprocal gamma function of the Laplace transform. Formally, it can be defined as ν ( x ) ≡ ∫ 0 ∞ x t d t Γ ( t + 1 ) ν ( x , α ) ≡ ∫ 0 ∞ x α + t d t Γ ( α + t + 1 ) {\displaystyle {\begin{aligned}\nu (x)&\equiv \int _{0}^{\infty }{\frac {x^{t}\,dt}{\Gamma (t+1)}}\\[10pt]\nu (x,\alpha )&\equiv \int _{0}^{\infty }{\frac {x^{\alpha +t}\,dt}{\Gamma (\alpha +t+1)}}\end{aligned}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the Gamma function.