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Why $\int e^{-x^2} dx$ cannot be expressed in a closed form? Definite integration of this function is possible however I could not understand why indefinite Integration is not possible.

user464196
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  • What do wupu mean when you say that “Definite integration of this function is possible”? How do you do that? – José Carlos Santos Jun 02 '19 at 14:09
  • It is not that it is impossible, actually the indefinite integral exists, but cannot be expressed in terms of "elementary" (don't know the precise term) functions. – G Cab Jun 02 '19 at 14:09
  • There are complicated algorithms which allow us to tell if a certain function has an antiderivative expressible in terms of elementary functions. See here for details (Risch algorithm). It is well known that $e^{-x^2}$ isn't one of those functions. – Jakobian Jun 02 '19 at 14:12
  • As far as I know, the only way to integrate $\int e^{-x^2},dx$ is to use the infinite series representation for $e^{-x^2}$. But the answer is going to be another infinite series. – Michael Rybkin Jun 02 '19 at 14:14
  • Please look a glance on this link "http://math.hunter.cuny.edu/ksda/papers/rick_09_02.pdf" Page No. 20 (Corollary 4.4), you can find your answer. – nmasanta Jun 02 '19 at 16:40
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