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We know that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\int_0^\infty \frac{\sin x}{x} \, dx=\frac{\pi}{2}.$$

How do I show that $$\int_0^\infty \left\vert\frac{\sin x}{x} \right\vert \, dx$$ converges?

Michael Hardy
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3 Answers3

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It doesn't. Using the convexity of $1/x$,

$$\int_0^\infty \left\vert\frac{\sin x}{x}\right\vert \,\mathrm{d}x=\sum_{k=0}^\infty\int_{k\pi}^{(k+1)\pi}\left\vert\frac{\sin x}{x}\right\vert \,\mathrm{d}x>\sum_{k=0}^\infty\int_{k\pi}^{(k+1)\pi}\frac{\left\vert\sin x\right\vert}{(k+1/2)\pi} \,\mathrm{d}x=\frac{2}{\pi}\sum_{k=0}^\infty\frac{1}{k+1/2}\;,$$

which diverges since the harmonic series diverges.

joriki
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  • @user7815: You're welcome! – joriki Mar 07 '11 at 20:08
  • I don't see how you do it using the convexity of $1/x$. You can take in the denominator $k\pi$ since $1/x$ is decreasing. – Bogdan Mar 07 '11 at 20:09
  • @Mielu: You're right, that would have been easier :-) I'm using the convexity by replacing the argument of $1/x$ in the sum of the contributions at $(k+1/2)\pi-\Delta$ and $(k+1/2)\pi+\Delta$ (which have the same value of $\lvert\sin x\rvert$) by the average value $(k+1/2)\pi$. – joriki Mar 07 '11 at 20:13
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It doesn't. This function is a sequence of bumps of decreasing size. The $n^{\text{th}}$ "bump" bounds area on the order of $\frac{1}{n}$ (there are a number of ways to show this), but $\sum_{n=1}^\infty \frac{1}{n} = \infty$.

Noah Stein
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$\int_0^\infty \frac{\sin x}{x} dx$ is in fact my most favorite example of a function which is Riemann integrable but not Lesbegue integrable. (just to tease people which think that Lesbegue is so much better than Riemann; in fact this integral might appear more frequent in applications than any integral which is Lesbegue but not Riemann integrable)

Fabian
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    In defense of your teasing victims, if you allow improper Riemann integrals, then why not improper Lebesgue integrals? The limit of Lebesgue integrals $\lim_{b\to \infty}\int_0^b\frac{\sin x}{x}$ is just the same as that for the Riemann integrals, which after all is what this "improper" integral means. But I do agree that it is a good example, not just for teasing people, but also to clarify the differences in the definitions of improper Riemann integrals versus ordinary Lebesgue integrals on unbounded domains. – Jonas Meyer Mar 07 '11 at 20:01
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    I do hope that your teasing victims have no delusions that application to functions in the real world is the reason that Lebesgue integration is so much better than Riemann integration. Anyway, +1. – Jonas Meyer Mar 07 '11 at 20:05
  • @Meyer: I agree and I don't want to give the impression that I don't see the beauty of Lesbegues (e.g., the dominated convergence theorem), but Lesbegues doesn't solve all problems and the sinc integral stays an improper integral... – Fabian Mar 07 '11 at 21:00
  • Well said. Your example is used in Gelbaum & Olmsted's book "Counterexamples in Analysis" for a function that is Riemann integrable but not Lebesgue integrable, and was the first time I saw it. Although I too tend to favor Lebesgue integration over Riemann (due to its more generalized nature), I still very much enjoy counterexamples such as this, as a reminder that things in Mathematics can often be much more interesting, than my knee-jerk assumptions might lead me. – Jonathan Hoyle Dec 19 '22 at 17:44