A family of integrals analytically solvable

Pedro Pagola (1) , Jose Luis Lopez (1)

1 UPNA - Universidad Pública de Navarra [Espagne] = Public University of Navarra

We consider this family of definite integrals: $$\int_{0}^{\infty} \frac{t^n}{(t-\lambda)^k}\,e^{-x t^m}dt, \hskip 0.4cm \Re x>0\hskip 0.15cm,\hskip 0.15cm m,k\in \mathbb{N}\hskip 0.15cm,\hskip 0.15cm \lambda\in\mathbb{C}\setminus\{\mathbb{R}^+\cup 0\}$$ Its explicit computation cannot be found in the traditional table books for general values of the parameters. It cannot be computed by algebraic manipulators either. Mathematica 7.0 computes the integral for some particular (low) values of m. In this paper we compute this family of integrals for general values of the parameters in terms of well known special functions: Gamma and Exponential integrals. From this computation, it is then straightforward to design a code for its evaluation with Mathematica and incorporate those integrals to the Mathematica library of integrals. We've compared the computation's speed of that integral for m = 1, 3 by using our code on the one hand and the command Integrate of Mathematica 7.0 on the other hand, concluding that our code is extraordinary faster.

Mots clés: Physical Sciences

Domaines: Informatique [cs]

Pedro Pagola, Jose Luis Lopez. A family of integrals analytically solvable. International Journal of Computer Mathematics, 2011, pp.1. ⟨10.1080/00207160.2011.554542⟩. ⟨hal-00707581⟩