I'm looking for various ways to evaluate the integral: $$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$

I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but I'm having trouble constructing the integrand properly. Perhaps we have to take into account that the root here may cause some trouble and define a function having a branch and this complicates things.

I know two solutions using real analysis. One uses Laplace transformations and the other using only elementary tools plus the known results of the Fresnel Integrals.

Can someone help me with the contour integration? Thank you!

  • $\begingroup$ The integral is undefined. $\endgroup$ – science Mar 8 '15 at 21:42
  • $\begingroup$ I agree with @science, as written the integrand function is not Riemann integrable over $\mathbb{R}$^+. Are you sure it is $dx$ and not $\frac{dx}{\sqrt{x}}$ or $\frac{dx}{x}$? $\endgroup$ – Jack D'Aurizio Mar 8 '15 at 21:47
  • $\begingroup$ What do you mean undefined? Yep, I'm completely sure about that. $\endgroup$ – Tolaso Mar 8 '15 at 21:49
  • $\begingroup$ The result of the above integral is $\displaystyle \frac{\sqrt{\pi}}{2}\sin\bigl({\tfrac{3\pi-1}{4}}\bigl)$. And I can assure you it converges. $\endgroup$ – Tolaso Mar 8 '15 at 21:54
  • 2
    $\begingroup$ I can assure you it converges. - And I can assure you that it does not, but rather oscillates indefinitely within the limits of a specified interval of the form $(a-1,a+1)$, where $$a=\dfrac{\sqrt{2\pi}}4\bigg(\cos\dfrac14+\sin\dfrac14\bigg).$$ $\endgroup$ – Lucian Mar 8 '15 at 23:04

I'm posting an asnwer (of the $2$ I have) using real analysis methods:

$$\begin{aligned} \int_{0}^{\infty}\sin x \sin \sqrt{x}\,dx &\overset{\sqrt{x}=u}{=\! =\! =\!}2\int_{0}^{\infty}u\sin u \sin u^2 \,du \\ &=-\int_{0}^{\infty}u\cos \left ( u^2+u \right )\,du+\int_{0}^{\infty}u\cos(u^2-u)\,du \\ &\overset{u \mapsto u+1}{=\! =\! =\! =\!}-\int_{0}^{\infty}u\cos(u^2+u)\,du+\int_{-1}^{\infty}\left ( u+1 \right )\cos\left ( u^2+u \right )\,du \\ &= \int_{0}^{\infty}\cos\left ( u^2+u \right )\,du+\int_{-1}^{0}\left ( u+1 \right )\cos\left ( u^2+u \right )\,du\\ &\overset{u={\rm v}-\frac{1}{2}}{=\! =\! =\! =\!}\int_{1/2}^{\infty}\cos\left ( {\rm v}^2-\frac{1}{4} \right )\,d{\rm v}+\int_{-1/2}^{1/2}\left ( {\rm v}+\frac{1}{2} \right )\cos \left ( {\rm v}^2-\frac{1}{4} \right )\,d{\rm v} \\ &= \int_{0}^{\infty}\cos \left ( {\rm v}^2-\frac{1}{4} \right )\,d{\rm v}+\\ & \left [ \int_{-1/2}^{1/2}{\rm v}\cos \left ( {\rm v}^2-\frac{1}{4} \right )\,d{\rm v}+\frac{1}{2}\int_{-1/2}^{0}\cos\left ( {\rm v}^2-\frac{1}{2} \right )\,d{\rm v}- \frac{1}{2}\int_{0}^{1/2}\cos\left ( {\rm v}^2-\frac{1}{2} \right )\,d{\rm v} \right ] \end{aligned}$$

However, the equation in the bracket equals zero due to symmetry.

Hence: $$\begin{aligned}\int_{0}^{\infty}\sin x \sin x^2\,dx&=\int_{0}^{\infty}\cos\left ( {\rm v}^2-\frac{1}{2} \right )\,d{\rm v}\\ &=\cos \frac{1}{4}\int_{0}^{\infty}\cos {\rm v}^2\,d{\rm v}+\sin \frac{1}{4}\int_{0}^{\infty}\sin {\rm v}^2\,d{\rm v}\\ &\overset{(*)}{=}\int_{0}^{\infty}\sin {\rm v}^2\,d{\rm v}\left ( \cos \frac{1}{4}+\sin \frac{1}{4} \right )\\ &=\frac{\sqrt{\pi}}{2}\sin \left ( \frac{3\pi-1}{4} \right )\;\; \;\;\;\; \square \end{aligned}$$

$(*)$ We used the Frensel integrals stating that $\displaystyle \int_{0}^{\infty}\cos x^2 \,dx=\int_{0}^{\infty}\sin x^2 \,dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$.

  • $\begingroup$ Thnaks for the nice answer. $+1$ $\endgroup$ – Claude Leibovici Mar 10 '15 at 11:10
  • $\begingroup$ You begin by writing the original integral as the difference of two others, both of which diverge. So I assume that the integral converges by means of the Cauchy principal value of its particular alternate expression. $\endgroup$ – Lucian Mar 11 '15 at 20:26

This is not an answer since it does not use complex analysis.

Using a CAS, what was found is that if $$I=\int_0^a \sin x\sin \sqrt{x}\,dx$$ $$I=\frac{1}{2} \sqrt{\frac{\pi }{2}} \left(\cos \left(\frac{1}{4}\right) (C(\alpha )+C(\beta ))+\sin \left(\frac{1}{4}\right) (S(\alpha )+S(\beta ))\right)-\sin \left(\sqrt{a}\right) \cos (a)$$ where appear Fresnel integrals using $\alpha=\frac{2 \sqrt{a}-1}{\sqrt{2 \pi }}$, $\beta=\frac{2 \sqrt{a}+1}{\sqrt{2 \pi }}$ which oscillates for ever as commented by Lucian.


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