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Thread: Integrable C-infinity function

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

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

    I found a function:

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

    which fits everything except it's only class $\displaystyle C^1$ (because $\displaystyle f''(1)$ 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
    MHF Contributor
    Opalg's Avatar
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    What about $\displaystyle e^{-x}/\sqrt x$? It is $\displaystyle C^\infty$, it goes to ∞ at x=0, it's integrable on (0,1] (by comparison with $\displaystyle 1/\sqrt x$), and it's integrable on [1,∞) (by comparison with $\displaystyle e^{-x}$).
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  3. #3
    Super Member redsoxfan325's Avatar
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    Excellent. I believe $\displaystyle \int_0^{\infty}\frac{e^{-x}}{\sqrt{x}}\,dx=\sqrt{\pi}$, for those curious.
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