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Bug with Limit, Series and ProductLog ?


With Mathematica 5.2 Windows I obtain

In[1]:=Limit[ ProductLog[Exp[a/x]/x]-a/x,x->0]
Out[1]= -Log[a]

which seems correct. But, setting a=1, I get

In[2]:=Limit[ ProductLog[Exp[1/x]/x]-1/x,x->0]
Out[2]=-8

which is inconsistent with the previous result
(except if Log[1] is Infinity !).

Worse, with Series I get

In[3]:=Series[ ProductLog[Exp[a/x]/x]-a/x,{x,0,5}]

Out[3]=\!\(\*
  InterpretationBox[
    RowBox[{\(-\(a\/x\)\), "+", "Indeterminate", "+",
      InterpretationBox[\(O[x]\^6\),
        SeriesData[ x, 0, {}, -1, 6, 1],
        Editable->False]}],
    SeriesData[ x, 0, {
      Times[ -1, a], Indeterminate}, -1, 6, 1],
    Editable->False]\)

Setting a=1 in the Series gives a complex answer.

How can I workaround the problem and get the correct
expansion for In[3]?
Thanks,
D.


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