Primitive of exp (-x^2) has no Solution in Elementary Functions

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Theorem

The primitive:

$\displaystyle \int \map \exp {-x^2} \rd x$

cannot be expressed in terms of a finite number of elementary functions.


Proof


Historical Note

The proof that $\ds \int \map \exp {-x^2} \rd x$ cannot be expressed with a finite number of elementary functions was first proved by Joseph Liouville.


Sources