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Does $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it?

Should I use 'principle value' to do: $$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$

21rw
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    What you propose would be called a "principal value" of the integral. But in fact the integral does not converge in the more stringent sense that mathematicians normally use. – GEdgar May 18 '17 at 00:13
  • @GEdgar Thanks for pointing it out. I've edited the question. – 21rw May 18 '17 at 01:11

2 Answers2

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Does $\int_{-\infty}^\infty \sin (t)\,\mathrm{d}t$ converge?

No, for the limit to exist with limit $L$, then for every pair of sequences $a_n\to-\infty$ and $b_n\to\infty$, we would have that

$$\lim\limits_{n\to\infty} \int_{a_n}^{b_n} \sin (t)\,\mathrm{d}t = L$$ which is obviously not the case.

The Cauchy Principal Value is different to usual convergence and this value does exist, it is $$\lim\limits_{a\to\infty}\int_{-a}^a\sin (t) \,\mathrm{d} t = 0.$$

Eff
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$$\int_{-a}^a \sin(t)\,dt=0$$ for $a>0$ since sine is an odd function. Hence

$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt=0.$$

Michael Hardy
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Sri-Amirthan Theivendran
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  • $a>0$ is not required. e.g., $\int_\pi^{-\pi} \sin t , dt = 0$, etc. –  May 18 '17 at 00:36