How can it be proven that the Gaussian function has no "trivial" primitives?

Asked Aug 28 '12 at 23:01

Active Aug 28 '12 at 23:22

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How can you prove that a function has no closed form integral?
Why can't erf be expressed in terms of elementary functions?

By trivial primitives I mean expressions involving logarithms, polynomials, and trigonometrical or exponential functions.

edited Apr 13 '17 at 12:19

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asked Aug 28 '12 at 23:01

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http://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory – Qiaochu Yuan Aug 28 '12 at 23:05
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Related: http://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral – cardinal Aug 28 '12 at 23:06
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Brian Conrad's Impossibility theorems for elementary integration is a very nice place to start, in my opinion. – cardinal Aug 28 '12 at 23:07
The answer to this question in general is found in the link that @cardinal supplied. This particular function is addressed in the link from Arybhata's comment on Eric Haengel's answer (here). Thus, I'm flagging to close as a duplicate. – mboratko Aug 28 '12 at 23:18

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