03-467 Leonardo F. Guidi, Domingos H. U. Marchetti

Convergence of Mayer Series via Cauchy-Kowalewski Majorant Methods with Application (412K, Postscript) Oct 14, 03

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Abstract. We construct majorant functions $\Phi (\beta ,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $% \vert z\vert <r(\beta )$ of convergence of Mayer series is given by the envelop of characteristics defined by the first crossing time of whole family. For non negative potentials we derive an explicit solution in terms of the Lambert $W$--function which is related to the exponential generating function of rooted trees $T$ as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we used this freedon to reexamine the ultraviolet problem of Yukawa potential. We also apply Cauchy--Kowalevsky theorem in order to discuss the analytic continuation of $% \Phi (\beta ,z)$ to the complex $z$--plane.

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