Is this function integrable?

• January 2nd 2011, 09:28 PM
aukie
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

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

• January 3rd 2011, 06:15 PM
aukie
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!