[SOLVED]A question on asymptotic expansion for erf function.

Alan

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

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).