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

${\displaystyle {\begin{aligned}\nu (x)&\equiv \int _{0}^{\infty }{\frac {x^{t}\,dt}{\Gamma (t+1)}}\\[10pt]\nu (x,\alpha )&\equiv \int _{0}^{\infty }{\frac {x^{\alpha +t}\,dt}{\Gamma (\alpha +t+1)}}\end{aligned}}}$

where ${\displaystyle \Gamma (z)}$ is the Gamma function.¹²