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$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does.

I'm trying to understand why and would like some help in working towards a formal proof.

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This is a problem from Rudin's Principles of Mathematical Analysis (#6.13 in my/the latest edition) where he has you fill out his outline. – Tyler Feb 3 '12 at 0:39
Do you at least have a basis for this statement? A.k.a. do you know if it's true before you attempted to prove it? What evidence do you have? If you can come up with an informal proof or some sort of educated conjecture then that might help you get started. I'm not saying it's not true, but you have to convince yourself that it's true before formally proving it. – chharvey Feb 3 '12 at 4:26
This is called Fresnel integral. – Jack Dec 9 '12 at 16:31
by the alternated series test, $\int_a^\infty \sin(x f(x)) dx$ converges whenever $f(x)$ is non-decreasing and $\to +\infty$. – user1952009 May 18 at 17:43
@Jozef: you can also use the Abel-Dirichlet's criterion. Appently, according to a comment to this similar question, you could use Riemann-Lebesgue lemma. – Watson Jul 25 at 10:14
up vote 30 down vote accepted

$x\mapsto \sin(x^2)$ is integrable on $[0,1]$, so we have to show that $\lim_{A\to +\infty}\int_1^A\sin(x^2)dx$ exists. Make the substitution $t=x^2$, then $x=\sqrt t$ and $dx=\frac{dt}{2\sqrt t}$. We have $$\int_1^A\sin(x^2)dx=\int_1^{A^2}\frac{\sin t}{2\sqrt t}dt=-\frac{\cos A^2}{2\sqrt A}+\frac{\cos 1}2+\frac 12\int_1^{A^2}\cos t\cdot t^{-3/2}\frac{-1}2dt,$$ and since $\lim_{A\to +\infty}-\frac{\cos A^2}{2\sqrt A}+\frac{\cos 1}2=\frac{\cos 1}2$ and the integral $\int_1^{+\infty}t^{-3/2}dt$ exists (is finite), we conclude that $\int_1^{+\infty}\sin(x^2)dx$ and so does $\int_0^{+\infty}\sin(x^2)dx$. This integral is computable thanks to the residues theorem.

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See what magic one can do when one does calculus carefully?One doesn't need fancy schmancy function spaces and measures all the time to do real math-sometimes all you have to do is know the basics and think it through carefully. : ) – Mathemagician1234 Feb 3 '12 at 3:36
$\int_0^c \frac{\sin t}{\sqrt t} \mathrm{d}t= - \int_1^c \frac{\mathrm{d} \cos t}{\sqrt t} = - \frac{\cos t}{\sqrt t}\big|_1^c + \int_1^c \cos t (-1/2t^{-3/2}) \mathrm{d}t = -\frac{\cos c}{\sqrt c} + \cos 1 - 1/2 \int_1^c \cos t t^{-3/2} \mathrm{d}t$ – RHS May 1 '13 at 5:57
"and the integral $\int_1^{+\infty}t^{-3/2}dt$ exists (is finite)" - where is the $\cos t$? – Elimination Jun 7 '15 at 14:17
@Elimination It prove that $\int_1^\infty|\cos t| t^{-3/2}dt$ converges. – Davide Giraudo Jun 7 '15 at 14:54

The humps for $x\mapsto \sin(x^2)$ go up and down. Each has an area smaller than that of the last. The areas converge to 0 as you progress down the $x$-axis. By the alternating series test, this converges.

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+1. No need to make things overly complicated. – Yonatan N Feb 3 '12 at 1:09

I solved this one integral as a particular case of the formula I provide here: under the name Weiler.

$$\int\limits_0^\infty {\sin \left( {a{x^2}} \right)\cos \left( {2bx} \right)dx} = \sqrt {\frac{\pi }{{8a}}} \left( {\cos \frac{{{b^2}}}{a} - \sin \frac{{{b^2}}}{a}} \right)$$

$$\int\limits_0^\infty {\cos \left( {a{x^2}} \right)\cos \left( {2bx} \right)dx} = \sqrt {\frac{\pi }{{8a}}} \left( {\cos \frac{{{b^2}}}{a} + \sin \frac{{{b^2}}}{a}} \right)$$

So you have

$$\int\limits_0^\infty {\sin \left( {{x^2}} \right)dx} = \sqrt {\frac{\pi }{8}} $$

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This is also informative, and works when there is no aspect with closed form. Taking Davide's substitution, define $$ A_n^+ = \int_{2 \pi n}^{2 \pi n + \pi} \; \frac{\sin t}{2 \sqrt t} \; dt \; , $$ $$ A_n^- = \int_{2 \pi n + \pi}^{2 \pi n + 2 \pi} \; \frac{\sin t}{2 \sqrt t} \; dt \; , $$ and finally $$ A_n = A_n^+ + A_n^- = \int_{2 \pi n }^{2 \pi n + 2 \pi} \; \frac{\sin t}{2 \sqrt t} \; dt \; , $$

Next, I just used $\int_{m \pi}^{m \pi + \pi} \sin t dt = \pm 2,$ depending upon the integer $m,$ and took bounds based on the size of the denominators.

I suppose we need to start with $n \geq 1.$ With that, $$ \frac{1}{\sqrt{2 \pi n + \pi}} \leq A_n^+ \leq \frac{1}{\sqrt{2 \pi n}}, $$ $$ \frac{-1}{\sqrt{2 \pi n + \pi}} \leq A_n^- \leq \frac{-1}{\sqrt{2 \pi n + 2 \pi}}, $$ and $$ 0 \leq A_n \leq \frac{1}{\sqrt{ 8 \pi} \; \; n^{3/2}}.$$

What does this say about convergence? The integral is $$ \sum_{n = 0}^\infty A_n. $$ Convergence does not depend on the initial terms, so we may start at the more convenient $n=1.$ From the $3/2$ exponent in the estimate of $A_n,$ we see that the sum is a finite constant. We do see modest oscillation in the indefinite integral, however the $\sqrt n$ terms in the denominators of $A_n^+$ and $A_n^-$ tell us that eventually the indefinite integral stays within any desired distance of the infinite integral.

This idea, cancellation of alternating contributions, can be used with far worse integrands, $\sin (x^5 - x - 1)$ comes to mind.

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There are more involved methods to actually compute this integral. A real variables method is given in this answer and a contour integration method is given in this answer.

Here is a much simpler method to show only the convergence of the integral. $$ \begin{align} \int_0^\infty\sin\left(x^2\right)\,\mathrm{d}x &=\int_0^\infty\frac{\sin(x)}{2\sqrt{x}}\,\mathrm{d}x\tag{1}\\ &=\sum_{k=0}^\infty\int_{2k\pi}^{(2k+2)\pi}\frac{\sin(x)}{2\sqrt{x}}\,\mathrm{d}x\tag{2}\\ &=\sum_{k=0}^\infty\int_{2k\pi}^{(2k+1)\pi}\frac{\sin(x)}2\left(\frac1{\sqrt{x}}-\frac1{\sqrt{x+\pi}}\right)\mathrm{d}x\tag{3}\\ &=\sum_{k=0}^\infty\int_{2k\pi}^{(2k+1)\pi}\frac{\sin(x)}2\frac{\pi}{\sqrt{x}\sqrt{x+\pi}\left(\sqrt{x}+\sqrt{x+\pi}\right)}\mathrm{d}x\tag{4}\\ &\le\int_0^\pi\frac{\sin(x)}2\left(1+\sum_{k=1}^\infty\frac1{4\sqrt{2\pi} k^{3/2}}\right)\mathrm{d}x\tag{5}\\ &=1+\frac{\zeta\!\left(\frac32\right)}{4\sqrt{2\pi}}\tag{6} \end{align} $$ Explanation:
$(1)$: substitute $x\mapsto\sqrt{x}$
$(2)$: break the integral into $2\pi$ segments
$(3)$: $\sin(x+\pi)=-\sin(x)$
$(4)$: algebra
$(5)$: $\frac{\pi}{\sqrt{x}\sqrt{x+\pi}\left(\sqrt{x}+\sqrt{x+\pi}\right)}\le\min\left(1,\frac1{4\sqrt{2\pi} k^{3/2}}\right)$ for $x\ge2k\pi$
$(6)$: evaluate integral and sum

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