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Math Help - could integration of exp(x^2) be solved?

  1. #1
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    could integration of exp(x^2) be solved?

    Hi every body

    I want to have the solution for the integration of exp(x^2)

    Thanks
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  2. #2
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    Quote Originally Posted by raladin View Post
    Hi every body

    I want to have the solution for the integration of exp(x^2)

    Thanks
    \int_0^{\infty} dx \, e^{x^2} has a known value, but the indefinite integral doesn't have a "closed form." That is to say, the indefinite integral can't be expressed as a finite sum, product, etc. of polynomials, trig functions, etc. I can't tell you how to prove that, but I know it has been proven.

    -Dan
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  3. #3
    TD!
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    The nice thing about mathematics is that when you encounter something, preferably something useful, but it isn't "possible" yet, you just define it.

    The integral of e(x) (usually with a minus-sign and some constants) appears to be very useful in e.g. statistics (normal distribution), but it doesn't have a closed-form primitive function, i.e. its primitive function cannot be expressed in terms of a finite composition of what we call 'elementary functions' (polynomials, trig, log/exp, ...).

    Here comes the mathematician: we just give it a new name, we define it as a new function (search for the error-function, Erf(x)).
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    The name of the math that deals with these problems is called "differencial algebra" (I know not a creative name).

    There is a theorem by Louiville:
    If f,g are rational functions and g is not a constant, then,
    \smallint f\exp (g) is elementary if and only if there exists a rational solution to the differencial equation,
    f=h'+rg
    On some open interval.
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    Quote Originally Posted by TD! View Post

    Here comes the mathematician: we just give it a new name, we define it as a new function (search for the error-function, Erf(x)).
    It reminds me of a joke. A mathemation was asked to capture sheep for a farmer in smallest amount of fencing possible. What does he do? Him builds a fence around himself and says "I declare this the outside" *)

    *)Hmmm... the Jordan Closed Curve theorem does not work here, interesting.
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