CARMA ANALYSIS SEMINAR Speaker: Prof David Jeffrey, Department of Applied Mathematics, University of Western Ontario Title: Investigations of some Stieltjes functions and some completely monotonic functions Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Tue, 21st Feb 2012 Abstract: Two sets of functions are studied to ascertain whether they are Stieltjes functions and whether they are completely monotonic. The first group of functions are all built from the Lambert $W$ function. The $W$ function will be reviewed briefly. It will be shown that $W$ is Bernstein and various functions containing $W$ are Stieltjes. Explicit expressions for the Stieltjes transforms are obtained. We also give some new results regarding general Stieltjes functions. The second set of functions were posed as a challenge by Christian Berg in 2002. The functions are $(1+a/x)^{(x+b)}$ for various $a$ and $b$. We show that the functions is Stieltjes for some ranges of $a,b$ and investigate experimentally complete monotonicity for a larger range. We claim an accurate experimental value for the range. My co-authors are Rob Corless, Peter Borwein, German Kalugin and Songxin Liang. [Permanent link]