[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.
Solve[y==Exp[x]-x,x] {{x -> -y - ProductLog[-E^(-y)]}} (* warning:: inverse functions... *) Normal[Series[-y - ProductLog[E^(-y)],{y,0,4}]/. ProductLog->W] = (-1 + 1/(1 + E^W[1]))*y + (-(1/(2*(1 + E^W[1]))) + (1 + 2*E^W[1])/(2*(1 + E^W[1])^3))*y^2 + (1/(6*(1 + E^W[1])) - (1 + 2*E^W[1])/(2*(1 + E^W[1])^3) + (2 + 8*E^W[1] + 9*E^(2*W[1]))/(6*(1 + E^W[1])^5))*y^3 + (-(1/(24*(1 + E^W[1]))) + (7*(1 + 2*E^W[1]))/(24*(1 + E^W[1])^3) - (2 + 8*E^W[1] + 9*E^(2*W[1]))/(4*(1 + E^W[1])^5) + (6 + 36*E^W[1] + 79*E^(2*W[1]) + 64*E^(3*W[1]))/(24*(1 + E^W[1])^7))* y^4 - W[1] using floating point: -0.56714329040978387299996866221035531973 - 0.63810374336511077852240738551988005289 *y - 0.0736778051763727578161965154457506468178701 *y^2 - 0.0013428596549900872478388497520605370350877 *y^3 + 0.0016360651479124971778419453788130981674683 *y^4 with N[ProductLog[1], 24]= 0.56714329040978387299996866221035531973 E^ % = 1.76322283435189671022520177695170665503 Wouter. ----- Original Message ----- From: "David Wilson" <davidwwilson@comcast.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, August 14, 2007 1:29 PM 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 __________ NOD32 2459 (20070814) Informatie __________ Dit bericht is gecontroleerd door het NOD32 Antivirus Systeem. http://www.nod32.nl
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
participants (3)
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David W. Cantrell -
David Wilson -
wouter meeussen