MHB A question on asymptotic expansion for erf function.

AI Thread Summary
The discussion focuses on finding an asymptotic expansion for the error function erf(x) as x approaches infinity. The user correctly identifies the integral representation of erf(x) and attempts to express it in terms of integrals that simplify the evaluation. They encounter difficulty in expanding the term e^{-1/ξ^2} due to the variable ξ approaching zero, which complicates the power series expansion. The user suggests that integration by parts may be necessary to progress further. The conversation highlights the challenges of asymptotic analysis in mathematical functions.
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I've got the next question which I just want to see if I got it right, and if not then do correct me.

we have [math]erf(x)=\frac{2}{\sqrt \pi} \int_{0}^{x} e^{-t^2} dt[/math]

and I want to find an asymptotic expansion of this function when [math]x\rightarrow \infty[/math].

So here's what I have done:

[math]erf(x)=\frac{2}{\sqrt \pi} \int_{0}^{\infty} e^{-t^2}dt - \int_{x}^{\infty} e^{-t^2} dt = 1 +\int_{\frac{1}{x}}^{0} e^{-\frac{1}{\xi ^2}}\frac{d\xi}{\xi ^2}[/math]

where in my last step I used the next change of variables: [math]\xi=\frac{1}{t}[/math].
Now, I am kind of stuck here, I mean [math]\xi[/math] is smaller than [math]\frac{1}{x}\rightarrow 0[/math], but I cannot expand [math]e^{-\frac{1}{\xi ^2}}[/math] with a power seris in [math]\frac{1}{\xi^2}[/math] so what to do now?

Thanks (the question appears in Murray's asymptotic anaysis page 27, question 2).
 
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