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**1 - 2**of**2**### Kragujevac J. Math. 31 (2008) 17–27. THE SPECIAL FUNCTION , III.

, 2006

"... Abstract. We prove the following exact symbolic formula of the special function , in the entire s-complex plane with the negative real axis (including the origin) removed, with a double Laplace transform: (s) = L { 2 · δ (t) + L { 12pii · [ (t · e −ipi) − (t · e ipi)]}} where δ (t) stands for ..."

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Abstract. We prove the following exact symbolic formula of the special function , in the entire s-complex plane with the negative real axis (including the origin) removed, with a double Laplace transform: (s) = L { 2 · δ (t) + L { 12pii · [ (t · e −ipi) − (t · e ipi)]}} where δ (t) stands for the distribution of Dirac and e represents the Euler’s number. 1. THE EXACT SYMBOLIC FORMULA In [4] and [5] the author introduces and characterizes the discrete and special function in the complex field. We recall that the explicit formula of the special function , in the discrete field, is: [k,Ω (I`)] =

### Kragujevac J. Math. 29 (2006) 141–150. THE SPECIAL FUNCTION , II

, 2005

"... Abstract. We describe a method for estimating the special function , in the complex cut plane A = C \ (−∞, 0], with a Stieltjes transform, which implies that the function is logarithmically completely monotonic. To be complete, we find a nearly exact integral representation. At the end, we also e ..."

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Abstract. We describe a method for estimating the special function , in the complex cut plane A = C \ (−∞, 0], with a Stieltjes transform, which implies that the function is logarithmically completely monotonic. To be complete, we find a nearly exact integral representation. At the end, we also establish that 1 / (x) is a complete Bernstein function and we give the representation formula which is analogous to the Lévy-Khinchin formula. 1.