# Evaluation of an integral

1. Apr 29, 2005

### metrictensor

Does anyone know how to go about solving

$\int e^{(x^{2})} dx$

2. Apr 29, 2005

### Data

not expressible in terms of elementary functions. Using the "special" function, $\mbox{erf} (x)$ (the "error function", defined by $\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt$), you can express the integral in this way, though:

$$\int e^{(x^2)} \ dx = -\frac{i\sqrt{\pi}}{2}\mbox{erf}(ix)+C.$$

Essentially what this means is that you either have to compute definite integrals with this integrand numerically, or look up values on tables for the error function.

3. Apr 29, 2005

### metrictensor

This is helpful. I have the bounds. Thaks. It is weird that there is no analytical solution to something that looks so simple. I did write a taylor series and integrated that to get an infinite sum that is equal to the integral. It is:

$\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)n!}}$

4. Apr 29, 2005

### trancefishy

that's like the classic example of something that has no elementary solution...

5. Apr 30, 2005

### dextercioby

The "erf" function is tabulated for real arguments only...

Daniel.

6. Apr 30, 2005

### Data

7. Apr 30, 2005

### dextercioby

Give me a link to the page in A & Stegun where the erf function of complex arg is tabulated.

Daniel.

8. Apr 30, 2005