I'm guessing that you actually want the series for the inverse at 1. And of course the real inverse relation is bivalued... Using Mathematica, we can avoid the Lambert W function using InverseSeries[Series[Exp[x] - x, {x, 0, 5}], y] gives Sqrt[2] (y-1)^(1/2) - (y-1)/3 + (y-1)^(3/2)/(9 Sqrt[2]) - 2/135 (y-1)^2 +... for the positive inverse, while -InverseSeries[Series[Exp[-x] + x, {x, 0, 5}], y] gives -Sqrt[2] (y-1)^(1/2) - (y-1)/3 - (y-1)^(3/2)/(9 Sqrt[2]) - 2/135 (y-1)^2 +... for the negative inverse. Now we should also be able to get the result using the Lambert W function, but it seems that Mathematica 5.2 has an error in the coefficient of the fourth term and thereafter. OTOH, I'm in a big rush right now, so maybe the mistake is mine instead: In[33]:= Series[-y - ProductLog[-E^(-y)], {y, 1, 2}, Assumptions -> y > 1] Out[33]= SeriesData[y, 1, {-Sqrt[2], -1/3, -1/(9*Sqrt[2]), -1/3, 17/(48*Sqrt[2])}, 1, 6, 2] David W. Cantrell ----- Original Message ----- From: "David Wilson" <davidwwilson@comcast.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, August 14, 2007 12:29 Subject: [math-fun] Tech question Can someone give me the a few terms of the power series of the inverse of exp(x)-x at 0? Exact coefficients would be preferred if rational. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun