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Math Help - Integrable C-infinity function

  1. #1
    Super Member redsoxfan325's Avatar
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    Integrable C-infinity function

    Is there a function f(x) of class C^{\infty} defined on (0,\infty) such that \lim_{x\to0}f(x)=\infty and \int_0^{\infty}f(x)\,dx exists and is finite?

    I found a function:

    f(x)=\left\{\begin{array}{lr}\frac{1}{\sqrt{x}}:&0  <x\leq1\\e^{\frac{1-x}{2}}:&1\leq x<\infty\end{array}\right\}

    which fits everything except it's only class C^1 (because f''(1) DNE).

    Any ideas? I'm sure such a function exists; I'm just not sure whether it can be expressed in terms of elementary functions.

    This is not a homework problem; I was just thinking.
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  2. #2
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    Opalg's Avatar
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    What about e^{-x}/\sqrt x? It is C^\infty, it goes to ∞ at x=0, it's integrable on (0,1] (by comparison with 1/\sqrt x), and it's integrable on [1,∞) (by comparison with e^{-x}).
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  3. #3
    Super Member redsoxfan325's Avatar
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    Excellent. I believe \int_0^{\infty}\frac{e^{-x}}{\sqrt{x}}\,dx=\sqrt{\pi}, for those curious.
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