1
$\begingroup$

How can I do a closed form expansion of $\int e^{x^2} \, dx$? Please be specific as to which method I must use.

$\endgroup$
  • $\begingroup$ The imaginary error function is defined as $$\text{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2}\,dt$$Then you can use the Taylor series or Burmann Series expansion. $\endgroup$ – Mark Viola Dec 1 '15 at 22:39
  • $\begingroup$ It's not an elementary function - it cannot be expressed using rational functions, radicals, exponential/logarithmic or trigonometric functions or any composition thereof. So we simply gave it a name. $\endgroup$ – whacka Dec 1 '15 at 22:43
  • $\begingroup$ This was proven by Liouville.............. $\endgroup$ – DanielWainfleet Dec 2 '15 at 2:23
1
$\begingroup$

$\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{-\infty}^x e^{-t^2} \ dt$ is known (at least after multiplication by a constant) as the Error function. There is no simple closed form for the Error function in terms of elementary functions such as polynomial, exponential or trigonometric functions.

Through a little manipulation, you can show that $\frac{2}{\sqrt{\pi}}\int_{0}^x e^{t^2} \ dt = i\mbox{ erf}(ix)$, which is known as the imaginary Error function. Clearly, since it's just a transformation of the normal Error function, it has no nice closed form either.

$\endgroup$
  • 1
    $\begingroup$ Not just "no simple closed form". It is provably not an elementary function. $\endgroup$ – Robert Israel Dec 1 '15 at 22:46
  • $\begingroup$ See "Elementary function" on wikipedia. $\endgroup$ – DanielWainfleet Dec 2 '15 at 2:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.