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Nu function

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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)}}\\\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.

See also

References

  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.
  2. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) . Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8th ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.

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