# A question on asymptotic expansion for erf function.

• MHB
• Alone
In summary, the conversation discusses finding an asymptotic expansion of the function erf(x) as x approaches infinity. The speaker has already made some progress in their approach by using a change of variables and integration by parts. However, they are currently stuck because they cannot expand e^{-\frac{1}{\xi^2}} with a power series in \frac{1}{\xi^2}. They mention a paper by Murray's asymptotic analysis that may provide some guidance for their problem.
Alone
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).

## What is the definition of the error function (erf)?

The error function (erf) is a special mathematical function that is used to calculate the probability of a normally distributed random variable falling within a certain range of values.

## What is an asymptotic expansion for the erf function?

An asymptotic expansion for the erf function is a mathematical representation of the function as a series of polynomial terms that can be used to approximate the value of the function for large input values.

## Why is an asymptotic expansion useful for the erf function?

An asymptotic expansion for the erf function is useful because it allows for a more efficient and accurate way to calculate the value of the function for large input values, which may be difficult to compute using the original definition of the function.

## What is the order of the polynomial used in an asymptotic expansion for the erf function?

The order of the polynomial used in an asymptotic expansion for the erf function is typically chosen based on the desired level of accuracy and can vary depending on the specific application.

## How is an asymptotic expansion for the erf function derived?

An asymptotic expansion for the erf function is derived using techniques from calculus and analysis, such as Taylor series and limits, to approximate the original function with a series of polynomial terms that become increasingly accurate as the input value increases.

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