# Approximations to the guassian integral

1. Oct 20, 2007

### Euclid

I assume this is standard but I wouldn't know where to look. How do you approximate an integral of the form
$$\int_{x_1}^{x_2}\exp(-x^2)dx$$
where possibly one of the endpoints is infinite?

2. Oct 20, 2007

### Gib Z

Taking into account that the integrand is an even function, and the well known result that :$$\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}$$ It remains only to calculate a integral of the form:

$$\int_0^x e^{-t^2} dt$$.

You may calculate the taylor series and integrate term by term to get a series for that integral: $$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{n! (2n+1)} =\left(x-\frac{x^3}{3}+\frac{x^5}{10}-\frac{x^7}{42}+\frac{x^9}{216}-\ \cdots\right)$$,

Or alternatively, the Error function ( Erf(x) ) is only a constant multiple of that integral, and its values are well tabulated on the internet. http://en.wikipedia.org/wiki/Error_function

3. Oct 21, 2007

### Euclid

Ah... thanks. That is very useful.