# Nu function

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Mathematical function

In mathematics, the **nu function** is a generalization of the [reciprocal gamma function](/source/Reciprocal_gamma_function) of the [Laplace transform](/source/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](/source/Gamma_function).[1][2]

## See also

## References

1. **[^](#cite_ref-1)** Erdélyi, A; Magnus, W; Tricomi, FG; Oberhettinger, F (1981). *Higher Transcendental Functions, Vol. 3: The Function*y*(*x*) and Related Functions*. pp. 217–224.

1. **[^](#cite_ref-Zwillinger_2014_2-0)** [Gradshteyn, Izrail Solomonovich](/source/Izrail_Solomonovich_Gradshteyn); [Ryzhik, Iosif Moiseevich](/source/Iosif_Moiseevich_Ryzhik); [Geronimus, Yuri Veniaminovich](/source/Yuri_Veniaminovich_Geronimus); [Tseytlin, Michail Yulyevich](/source/Michail_Yulyevich_Tseytlin); Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; [Moll, Victor Hugo](/source/Victor_Hugo_Moll) (eds.). [*Table of Integrals, Series, and Products*](/source/Gradshteyn_and_Ryzhik). Translated by Scripta Technica, Inc. (8th ed.). [Academic Press, Inc.](/source/Academic_Press%2C_Inc.) [ISBN](/source/ISBN_(identifier)) [978-0-12-384933-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-384933-5). [LCCN](/source/LCCN_(identifier)) [2014010276](https://lccn.loc.gov/2014010276).

## External links

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Nu function"](https://mathworld.wolfram.com/NuFunction.html). *[MathWorld](/source/MathWorld)*.

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