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A long time ago I heard somethings about the following primitive $\int e^{x^2} \ dx$ :

1 ) It doens't exist

2) It cannot be expressed by means of "elementary functions"

Someone could point me a reference about the existence (if possible a reference that treats about the explicit form) of this primitive? What is the precise meaning of the term "elementary functions" ?

Thanks for your attention

math student
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  • Certainly this function has a primitive, namely, $\int_{x_0}^x e^{t^2} dt$, for example (any other differs from this one by a constant). As for "elementary functions", see this popular question: http://math.stackexchange.com/questions/118113/what-makes-elementary-functions-elementary – Travis Willse Jan 10 '15 at 03:57
  • But this integral have an "explicit" form ? – math student Jan 10 '15 at 03:58
  • Do you mean one not involving a definite integral? In that case, it depends on which functions you allow. One can write a primitive in terms of the Dawson function---http://en.wikipedia.org/wiki/Dawson_function---but that's more or less a tautology, as it's usually defined to be (a modification of) that primitive in the first place. – Travis Willse Jan 10 '15 at 04:21

1 Answers1

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This just mean that it cannot be expressed in terms of elementary functions like $$ 5x^{3}-2x+3, \log[x], e^{x}, \sin[x],5^{x}\cdots $$ which are the ones we encountered in a typical calculus I class. The typical function of the type you mentioned is $$ \frac{\sin[x]}{x} $$ And it is not too difficult to change your integral to something close to this form.

The sub-branch that deal with whether an explicit integral can be obtained using elementary functions is called differential algebra. The major tool is Liouville's theorem. Regrettably, the founder of the subject, Joseph Ritt committed suicide despite his monumental mathematical work.

Bombyx mori
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