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Question about integrals

  1. Dec 11, 2012 #1
    Is it possible to do Gaussian integrals with half integrals.
    we would define then nth derivative of [itex] e^{-x^2} [/itex]
    and then somehow use that. And this integral is over all space.
    any input will be much appreciated.
  2. jcsd
  3. Dec 11, 2012 #2


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    Science Advisor

    Your question is vague. What do you mean by?
  4. Dec 12, 2012 #3
    for example if we had to integrate [itex] e^{ax} [/itex] then nth derivative would be
    [itex] a^ne^{ax} [/itex] so the half dervative would be
    [itex] a^{.5}e^{ax} [/itex] and the half integral would be
    [itex] \frac{e^{ax}}{a^{.5}} [/itex]
    I was just wondering if we could use this to help us evaluate a Gaussian integral.
  5. Dec 12, 2012 #4


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    Homework Helper

    That is called fractional calculus
    Half integrals depend on arbitrary constants we might have for the half integral of e^(ax)
    sqrt(pi/a) e^(a x) erf(sqrt(a x))

    I would not be surprising that this could be used, but I am not sure it would be easier or more interesting than other popular methods.

    erf(x) function and gamma functions pop out all the time when taking half integrals and your integral is easily expressed in terms of them.

    Here is some stuff about all the fun ways to find the integral.
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