Re: Bug with Limit, Series and ProductLog ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg61443] Re: [mg61402] Bug with Limit, Series and ProductLog ?*From*: "Carl K. Woll" <carl at woll2woll.com>*Date*: Wed, 19 Oct 2005 02:16:38 -0400 (EDT)*References*: <200510180644.CAA11217@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

did wrote: > 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. > Here is one approach. Define f as your function: In[1]:= f[x_] := ProductLog[Exp[a/x]/x] - a/x Now, differentiate with respect to x: In[16]:= f'[x] Out[16]= a/x a/x a/x a E E E (-(------) - ----) x ProductLog[----] 3 2 x a x x -- + ------------------------------------- 2 a/x x a/x E E (1 + ProductLog[----]) x Notice the presence of the same ProductLog as exists in f. We can replace these occurrences of ProductLog with f[x]+a/x and obtain a differential equation for f. Since f already has a definition, I'll use g: In[18]:= g'[x] == f'[x] /. _ProductLog -> g[x] + a/x // Simplify Out[18]= g[x] g'[x] == -(--------------) a + x + x g[x] It's convenient to rewrite this equation as ode==0, where ode is: In[19]:= ode = Numerator[Together[g'[x] - f'[x] /. _ProductLog -> g[x] + a/x]] Out[19]= g[x] + a g'[x] + x g'[x] + x g[x] g'[x] Now, we simply find the series expansion of this expression and set all the terms to 0: In[23]:= sol = Solve[ode + O[x]^2 == 0, {g'[0], g''[0]}] Out[23]= g[0] (2 + g[0]) g[0] {{g''[0] -> ---------------, g'[0] -> -(----)}} 2 a a Plug this solution into your power series for g: In[26]:= g[x] + O[x]^3 /. sol[[1]] Out[26]= 2 g[0] x g[0] (2 + g[0]) x 3 g[0] - ------ + ------------------ + O[x] a 2 2 a Finally, use the value of g[0] you obtained: In[27]:= % /. g[0] -> -Log[a] Out[27]= 2 Log[a] x (2 - Log[a]) Log[a] x 3 -Log[a] + -------- - ---------------------- + O[x] a 2 2 a It's a simple matter to obtain even higher orders of the series simply by repeating the above procedure. I checked the results for various values of a and x, and it seems like the above expansion is correct. Carl Woll Wolfram Research

**References**:**Bug with Limit, Series and ProductLog ?***From:*"did" <didier.oslo@hotmail.com>