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Why can't be we take the antiderivative of $e^{y^2}$ with respect to y? My textbook just says it is not possible. That's it. No explanation. I could not find any one explanation on the web where it was explained either. Might seem like a dorky question but any explanation would be appreciated. But if we can, I guess the next natural question would be, "what is the anti-derivative?"

Eames Cobb
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    You absolutely can, there's just no closed form way of expressing it. – rurouniwallace Jul 27 '13 at 02:49
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    Well, we can, but the result is not an elementary function, that is, a combination of the standard functions that we know and love. – André Nicolas Jul 27 '13 at 02:49
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    What textbook is this? – RghtHndSd Jul 27 '13 at 02:56
  • The reason your book does not explain anything about the proof is that it really is beyond the level of a calculus textbook. For a quick sketch of the relevant mathematics, see this note by Matthew Wiener. This answer has additional references, and quite a few problems on this site are relevant as well. The key phrase is "elementary integration." – Andrés E. Caicedo Jul 27 '13 at 02:57
  • If you've had power series already, write out the general term for $ \ e^{x^2} \ $ and integrate it term-by-term. Does the resulting series resemble anything related to the elementary functions that you know? Instead, we simply define a new (non-elementary) function which is described by that series, as described by rghthndsd below. (In fact, many useful ("special") functions are defined by such integrals...) – colormegone Jul 27 '13 at 04:15

2 Answers2

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Brian Conrad has this article going through what it means to say an integral is "impossible", among other things. However it is rather on the advanced side.

As for the anti-derivative, it is defined to be $\frac{1}{2}\sqrt{\pi}\operatorname{erfi}(y)+C$. You can see a graph of this function at wolfram-alpha.

RghtHndSd
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  • In reality Brian Conrad article gives an "elementary" proof (modulo the language) of that impossibility. In particular he does not use differential Galois theory. I think in principle an ambitious freshmen could go through his proof (in reality of course some field theory would be nice). – blabler Jul 27 '13 at 03:15
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G. H. Hardy's The Integration of Functions of a Single Variable (second edition (1916), gratis download from Project Gutenberg) may be of interest as a readable exposition.

(He cites theorems of Liouville that imply $\int e^{x^2}\, dx$ cannot be expressed in elementary terms (pp. 62 ff.), so this reference is not a complete answer to the original question.)

Andrew D. Hwang
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