evaluation of an integral

[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!

http://www.amazon.com/exec/obidos/tg/detail/-/0486612724/qid=1114880468/sr=8-1/ref=pd_csp_1/002-0334771-5287233?v=glance&s=books&n=507846

: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.