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Domain of validity of an integral

  1. Jul 26, 2014 #1
    1. The problem statement, all variables and given/known data

    This integral is actually part of a question I had on an exam about analytic continuation. The integral is
    [tex] \int_0^1 dx \frac{x^n}{\sqrt{x^3+5}} [/tex]. The first part of the question is "what is its domain of validity for absolute convergence of the integral?" It then goes on to ask for a valid analytic continuation (I know how to do that part)


    2. Relevant equations

    none


    3. The attempt at a solution

    The answer key my professor provided states that "the integrand only blows up at x=0, if at all, in a finite range of integration. As x -> 0, the integrand is O([itex]x^n[/itex]), so there is absolute convergence if Real n > -1.

    I can easily how how this is the domain of validity assuming the integral blows up at 0, but I can't see how the integral would possibly blow up at 0. Plugging in zero for x gives [itex]\frac{0}{\sqrt{5}}[/itex]. I've tried putting the integral in other forms and I still don't see it. I taylor expanded the square root around x = 0 but that still doesn't cause it to blow up, nor does multiplying the top and bottom of the integral by the square root. I'm sure I'm just making some stupid calculus mistake somewhere...
     
  2. jcsd
  3. Jul 26, 2014 #2

    Ray Vickson

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    Did you not read the restriction Re(n) > -1? What happens if you take Re(n) = -1 (for example, n = -1)? What happens if you take Re(n) < -1, for example, n = -2?
     
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