Integrating the Exponential Function: Techniques and Resources

[metrictensor]

Does anyone know how to go about solving

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

 

[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.

 

[metrictensor]

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.

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:

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

 

[trancefishy]

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

 

[dextercioby]

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


Daniel.

 

[Data]

Really? I have a tabulation for complex arguments in front of me right now!



:tongue:

 

[dextercioby]

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

Daniel.

 

[dextercioby]

Yes,page 325.You're right.Abramowitz & Stegun has it.

Direct link.

http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=325&Submit=Go

Daniel.