# What is the closed form of $\int e^{x^2} \, dx$?

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

• 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. – Mark Viola Dec 1 '15 at 22:39
• 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. – whacka Dec 1 '15 at 22:43
• This was proven by Liouville.............. – DanielWainfleet Dec 2 '15 at 2:23

$\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{-\infty}^x e^{-t^2} \ dt$ 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.