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Asymptotic expansion instead of a power series expansion

Posted 10 years ago
3 Replies
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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 > 0
f := 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.

3 Replies
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 BY: Daniel Lichtblau
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/4
Would 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.
Is this what you are looking for?
Series[Sqrt[Pi] Exp[x^2] Erfc[x], {x, Infinity, 15}, Assumptions -> x > 0] // TraditionalForm

Sometimes 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

POSTED BY: Sam Carrettie
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