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Integrate e^(x^2) dx

This is what I have.

∫ex2dx=∑n=0∞x2n+1n!(2n+1) \displaystyle\int{ e^{x^2}} dx =\displaystyle\sum_{n=0}^{\infty} \frac{x^{2n+1}} {n!(2n+1)}

But WolframAlpha gives 12πerfi(x) \frac{1}{2} \sqrt{\pi} erfi(x)

I know I'm right, because WA also gives a series expansion of the integral which is equivalent to mine.. I'm just wondering how the two answers are the same, and what the hell the erfi function is. Anybody come across it before?
(edited 12 years ago)
Reply 1
Avatar for gff
gff
5
Wolfram Alpha is tricking you with a definition.

Even though, I can't tell you anything about any of these functions, I've met them in some very simple probability questions.

1. Error Function
2. Gaussian integral
3. PDF
4. Normal Distribution

What WA does, is just to replace your integral with the definition, since no elementary functions exist for it.
About the Tylor series expansion, I would guess that they are the approximation of the definite integral over the interval, since it cannot be evaluated in closed form.
(Of course the infinite series represent the exact integral, I meant that they may be used for approximation as well).
(edited 12 years ago)

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