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Evaluation of a definite integral

  1. Aug 22, 2014 #1

    dyn

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    I am looking at a solution to an question and I don't understand how the value of the following definite integral comes out to be zero ? The following function is evaluated from 0 to ∞ with r being the variable

    ## exp(-β^2r^2)r^nr^-1/(n-1)## That should read r raised to the power of (n-1)
    I presume when the value of ∞ is input the negative exponential turns out to be zero but I don't understand how inputting zero gives a value of zero ?
     
    Last edited: Aug 22, 2014
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  3. Aug 22, 2014 #2

    Simon Bridge

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    which? where?
    Did you mean to write: $$\int_0^\infty e^{-\beta^2r^2}r^{n-1}\;dr$$

    What do you think it should come out to?
    What is beta? What is "n"?
     
  4. Aug 22, 2014 #3

    dyn

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    You have got the function right but it also has a factor of 1/(n-1). This function was derived from integration by parts. It just needs evaluating with the values ∞ and 0. When I input 0 I have exp(0) which is 1 and 0 to the power (n-1) which I thought is also 1. But overall the answer should be zero
     
  5. Aug 23, 2014 #4

    Simon Bridge

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    Oh you mean you've done an integral and you have ended up at this stage: $$\frac{1}{n-1}\left[ e^{-\beta^2r^2} r^{n-1}\right]_0^\infty$$
    ... when you put ##r=0## into ##r^{(n-1)}## you get ##0^{n-1}## right?
    What is zero multiplied by itself? i.e. if n=3, then n-1=2 and ##0^2=0\times 0 = ?##
     
    Last edited: Aug 23, 2014
  6. Aug 23, 2014 #5

    dyn

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    Thanks for that. Somehow I was under the impression that zero raised to any power was 1 !
     
  7. Aug 23, 2014 #6

    Simon Bridge

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    You just got a rule stuck in your head, that's all.
    Happens to everyone ;)
     
  8. Aug 24, 2014 #7

    dyn

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    Thanks. I did get the wrong rule stuck in my head. But what would happen at r=o if n=1 then I would have zero to the power of zero which I believe is 1 ? I also have a zero on the denominator. Is it impossible to evaluate in this case ? Incidentally the question did state n≥2 so I am just asking for future reference.
     
  9. Aug 24, 2014 #8

    SteamKing

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    You should review the rules of exponentiation:

    http://en.wikipedia.org/wiki/Exponentiation

    The expression 0^0 (zero raised to the zeroth power) is considered, in general, an indeterminate form. 0^n is defined only for integers n > 0. There may be certain formulas which require that 0^0 be evaluated as equal to one, but these are exceptions.
     
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