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Integration by parts does not seem to work. I was wondering if this integral could be solved using a specific contour and applying for example Jordan's lemma?

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    This is the imaginary error function.2012-03-21
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    "Find" or "evaluate" is the right word here; "solve" is not. One solves equations; one solves problems. One evaluates expressions. In this case, one seeks a _value_, not a _solution_.2012-03-21
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    I think [this](http://memegenerator.net/instance/16722124) is what @MichaelHardy is trying to say...2012-03-21
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    Hmm I remember you have to square this integral and use spherical coordinates.2012-03-21
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    One can solve the problem of evaluating the integral, though :)2012-03-21
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    @MichaelHardy, I was just replicating the wording used across the board, and actually the title was even edited and is no longer my own.2012-03-21
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    One can solve the problem of evaluating the integral. But one _solves_ a _problem_ and _evaluates_ and integral. As for "across the board": yes it's true; this is a frequent misuse of terminology among the naive, and the naive are numerous.2012-03-21
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    Now your comment makes sense, I just reread myself for the first time, my double mistake. ;-)2012-03-22

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Elementary functions are generally considered to be functions generated by the identity function, constant functions, basic trig functions, inverse trig functions, exponential functions, logarithmic functions, the four arithmetic operations, and composition. It is well-known that while $\mathrm{e}^{x^2}$ has an antiderivative, that antiderivative is not among the elementary functions.

A name has been given to a similar function $F$, where $F(p)=\frac{2}{\sqrt{\pi}}\int_0^{p}\mathrm{e}^{-x^2}\,dx$. The standard name of this function is the error function, denoted $\operatorname{erf}$. Note the minus sign in the exponent. This explains the name that oenamen gives to your function in his comment.

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    Thanks for the answers Alex and Oenamen. The question can be closed.2012-03-21
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    @citrucel: You're welcome.2012-03-21