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

Theorem

The primitive:

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

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

Proof

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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

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)