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I am wondering if the graph of

$$f(x) = \int_{0}^{x}{e^{-t^2}}\,\mathrm{d}t$$

is possible to be written as some function without the Error Function, since it looks like an existing function when I try to plot it using an advanced calculator.

If this is not possible, why not?

amWhy
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Ensz
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    Up to a constant, this is the error function $\operatorname{erf}$, so it's not what you call "imaginary". What is $\sin(x)$ then? Is it also "imaginary"? The functions that we usually see and work with are called "elementary". It is a very, very, very small class of functions. You could add the error function to that class, if you want. Similarly as, e.g., $\sin$ or $\cos$ it can be defined by power series. – amsmath Mar 01 '21 at 17:29
  • @amsmath so if i apply tailor series thats how far ill get to some 'elementary' function? – Ensz Mar 01 '21 at 17:48
  • i would even appreciate if there exists some equation that is equivalent when plotted – Ensz Mar 01 '21 at 18:00
  • https://en.wikipedia.org/wiki/Error_function#Properties – amsmath Mar 01 '21 at 18:01
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    The choice to regard a symbolic expression as not needing simplification is largely an aesthetic one (although various systems of choices can be formalized and result in useful mathematics). For example, given $x$, how many digits of that integral do you want? And how is that different from asking how many digits of $\pi$ (presumably a "solved" expression) that you want? If $\sin(1)$ feels like a solved problem, and that integral does not, what is the difference? These are open-ended questions with no right or wrong answers. – leslie townes Mar 01 '21 at 18:03
  • The fact that it cannot belong to a broad class called "elementary functions" is not easily nor briefly proven. – DanielWainfleet Mar 01 '21 at 22:27
  • Related: https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral – Hans Lundmark Mar 02 '21 at 08:39

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