Evaluation of an integral

  • #1
117
1

Main Question or Discussion Point

Does anyone know how to go about solving

[itex]\int e^{(x^{2})} dx[/itex]
 

Answers and Replies

  • #2
998
0
not expressible in terms of elementary functions. Using the "special" function, [itex]\mbox{erf} (x)[/itex] (the "error function", defined by [itex]\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt[/itex]), you can express the integral in this way, though:

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

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
117
1
Data said:
not expressible in terms of elementary functions. Using the "special" function, [itex]\mbox{erf} (x)[/itex] (the "error function", defined by [itex]\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt[/itex]), you can express the integral in this way, though:

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

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:

[itex]
\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)n!}}
[/itex]
 
  • #4
that's like the classic example of something that has no elementary solution...
 
  • #5
dextercioby
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The "erf" function is tabulated for real arguments only...


Daniel.
 
  • #7
dextercioby
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Give me a link to the page in A & Stegun where the erf function of complex arg is tabulated.

Daniel.
 

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