# Asymptotic expansion instead of a power series expansion

Posted 10 years ago
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 With Mathematica, the Series function gives a power series expansion. But I am looking for a different type of expansion. Let me explain :\$Assumptions = Element[x, Reals] && x > 0f := Exp[x^2]*Erfc[x]Series[f, {x, 0, 5}]This gives the result :SeriesData[x, 0, {1, (-2) Pi^Rational[-1, 2], 1, Rational[-4, 3] Pi^Rational[-1, 2],Rational[1, 2], Rational[-8, 15] Pi^Rational[-1, 2]}, 0, 6, 1]But I am interested in getting an asymptotic expansion of the formĀ  given in Abrahamson and Stegun (as shown below). How can I get these type of expansions with Mathematica? I would appreciate any help that I can get.
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Posted 10 years ago
 To see the relevance of the argument restriction, try taking the series at -infinity. My guess is the A&S formulae are intended to hold in as general a region as possible, so they are in a sense giving a "sector" at complex infinity.I'm not understanding your question regardingĀ  the general form for expansion around infinity.
Posted 10 years ago
 Thanks a lot for your reply. Yes, this is the solution that I was looking for. As an extra bonus, I can see the use of the SeriesCoefficient command also. Is there any reason why the general form is not given by mathematica for expansion around Infinity?I have another question which does not primarily relate to Mathematica. In the fomulae from Abrahamson and Stegun, I see the following condition attached :| arg z| < 3*Pi/4Would appreciate it if you could explain the relevance of this condition. Does it become relevant only when z is a complex number? All the better if you coud do this with Mathematica.Thanks.
Posted 10 years ago
 Is this what you are looking for?Series[Sqrt[Pi] Exp[x^2] Erfc[x], {x, Infinity, 15}, Assumptions -> x > 0] // TraditionalFormSometimes general formulas will work too, but I could not get it to work around Infinity:SeriesCoefficient[Exp[x^2] Erfc[x], {x, 0, n}] // TraditionalForm