# How to proof $\int_{(0,\infty)} \int_{(0,\infty)} |\sin x| \,\, e^{-xy} \,\, dx \,\, dy < \infty$

How can I proof this?

$$\int_0^\infty \int_0^\infty |\sin x| \,\, e^{-xy} \,\, dx \,\, dy < \infty$$

I tried to say $$\int_0^\infty \int_0^\infty |\sin x| \,\, e^{-xy} \,\, dx \,\, dy \leq \int_0^\infty \int_0^\infty e^{-xy} \,\, dx \,\, dy$$ but that didn't help since the right side is $$\infty$$. All other ideas failed as well.

My "main idea" was to use Tonelli Fubini:

$$\int_0^\infty \int_0^\infty |\sin x| \,\, e^{-xy} \,\, dy \,\,dx = \int_0^{\infty} \frac{1}{x} |\sin(x)| \,\,dx$$

But I don't know how to continue here now. I tried again to say $$\int_0^{\infty} \frac{1}{x} |\sin(x)| \,\,dx \leq \int_0^{\infty} \frac{1}{x} \,\,dx$$ but the right side is $$\infty$$ again.

For Tonelli it would be also enough to check $$\int_0^\infty \int_0^\infty |\sin x| \,\, e^{-xy} \,\, dx \,\,dy$$ about finity, but here I get no better result.

• You need to show your attempt otherwise the post will be deleted Commented Dec 22, 2021 at 18:32
• "All other ideas" - what are those? A hint to try something more useful: write$$\int_0^\infty e^{-xy}|\sin x|dx=\sum_{n\ge0}\Im\left(\int_{n\pi}^{(n+1)\pi}e^{-x(y-i)}dx-\int_{(n+1)\pi}^{(n+2)\pi}e^{-x(y-i)}dx\right).$$
– J.G.
Commented Dec 22, 2021 at 18:36
• What is that big $I$ before the brackets? I don't think we've had to notation Commented Dec 22, 2021 at 18:40
• @LukasNiessen It means "imaginary part". Commented Dec 22, 2021 at 18:42
• @K.defaoite is right. You might have instead seen $\operatorname{Im}$.
– J.G.
Commented Dec 22, 2021 at 18:44

Note that the function $$(x,y)\mapsto\left|\sin x\right|e^{-xy}$$ is non-negative and measurable, so Tonelli Theorem is applicable. We have that $$\begin{eqnarray*} & & \int_{0}^{\infty}\int_{0}^{\infty}\left|\sin x\right|e^{-xy}dydx\\ & = & \int_{0}^{\infty}\left|\sin x\right|\left(-\frac{1}{x}e^{-xy}\mid_{y=0}^{y=\infty}\right)dx\\ & = & \int_{0}^{\infty}\left|\frac{\sin x}{x}\right|dx\\ & = & \infty. \end{eqnarray*}$$ To prove that $$\int_{0}^{\infty}\left|\frac{\sin x}{x}\right|dx=\infty$$, we note that $$|\sin x|\geq\frac{1}{2}$$ for $$x\in[2n\pi+\frac{\pi}{6},2n\pi+\frac{5\pi}{6}]$$. Therefore, $$\begin{eqnarray*} \int_{0}^{\infty}\left|\frac{\sin x}{x}\right|dx & \geq & \sum_{n=0}^{\infty}\int_{2n\pi+\frac{\pi}{6}}^{2n\pi+\frac{5\pi}{6}}\left|\frac{\sin x}{x}\right|dx\\ & \geq & \sum_{n=0}^{\infty}\frac{1}{2}\cdot\frac{1}{2n\pi+\frac{5\pi}{6}}\cdot(\frac{2\pi}{3})\\ & = & \infty. \end{eqnarray*}$$ It is impossible to prove that $$\int_{0}^{\infty}\int_{0}^{\infty}\left|\sin x\right|e^{-xy}dydx<\infty$$. Is there any typo ?