Antiderivative of $e^{-x²}$ and $e^{x²}$ expressed as non- elementary functions called error function , I'm sorry to ask this question probably it is a wild guess to ask about the obtained consequence in mathematics if $e^{-x²}$ and $e^{x²}$ expressed as Elementary function since it is widely used in mathematics ? and Are these consequences w'd be good ?
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1Nothing. Risch's algorithm proves the non-existence of elementary antiderivatives. – Parcly Taxel Feb 08 '18 at 14:50
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1See this highly upvoted Question and Answer How to determine with certainty that a function has no elementary antiderivative? – hardmath Feb 08 '18 at 15:00
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Liouville's theorem plus elementary antiderivatives prohibited by it = contradiction! – Martín-Blas Pérez Pinilla Feb 08 '18 at 15:08
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AFAIK, no important direct consequences.
However, as with many other problems, notably the solution of polynomial equations in radicals, the road to proving that there are no elementary expressions for the integral of $e^{x^2}$ and other such functions has led to a deeper understanding of the problem, given by Liouville's theorem and the Risch algorithm used in most symbolic software, and also to Differential Galois theory.
lhf
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If your question is about the operational consequences, the impact is probably neglectible: a few statistical formulas easier to establish, a few more integrals solvable analytically. Nothing simpler as regards numerical evaluation.
Regarding the changes to the theory, that would falsify an established theorem and be a loophole in maths, which is not really possible.