- #1

Alone

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

and I want to find an asymptotic expansion of this function when \(\displaystyle x\rightarrow \infty\).

So here's what I have done:

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

where in my last step I used the next change of variables: \(\displaystyle \xi=\frac{1}{t}\).

Now, I am kind of stuck here, I mean \(\displaystyle \xi\) is smaller than \(\displaystyle \frac{1}{x}\rightarrow 0\), but I cannot expand \(\displaystyle e^{-\frac{1}{\xi ^2}}\) with a power seris in \(\displaystyle \frac{1}{\xi^2}\) so what to do now?

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