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Math Help - Is this function integrable?

  1. #1
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    Is this function integrable?

    How can I show the following function is integrable, given that \phi \in L(\mathbb{R})??

    \frac{2\phi (x)(\cos (h x)-1)}{i x}, where i=\sqrt{-1}, and h is a fixed constant.

    i.e. I want to show

    \int_{-\infty }^{\infty } |\frac{2\phi (x)(\cos (h x)-1)}{i x}| \, dx<\infty

    any ideas on how I might go about this??
    Last edited by aukie; January 3rd 2011 at 06:59 PM.
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  2. #2
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    For anyone interested, I think the proof follows along the lines, for any fixed h, the function \frac{2(\cos (h x)-1)}{x} is continuous and goes to 0 as x goes to plus or minus infinity, thus it is bounded by say M, hence we have,

    \int_{-\infty }^{\infty } |\frac{2\phi (x)(\cos (h x)-1)}{i x}| \, dx < M \int_{-\infty }^{\infty } |\phi (x)| \, dx < \infty

    simple when u know how I guess!
    Last edited by aukie; January 3rd 2011 at 09:31 PM.
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