Question and Answers Forum

All Questions      Topic List

Vector Questions

Previous in All Question      Next in All Question      

Previous in Vector      Next in Vector      

Question Number 128314 by 676597498 last updated on 06/Jan/21

$${\int }_0^{\mathrm{\infty }}\frac{\sin \left(\mathrm{px}\right)}{\sqrt{\mathrm{x}}}\mathrm{dx}$$$$\mathrm{\forall }\mathrm{p}\in \mathbb{R}$$

Answered by Dwaipayan Shikari last updated on 06/Jan/21

$${\int }_0^{\mathrm{\infty }}\frac{sinpx}{\sqrt{x}}dx=2{\int }_0^{\mathrm{\infty }}{sinpt}^{2}dtx=t^2$$$$=\frac{1}{i}{\int }_0^{\mathrm{\infty }}{e}^{{ipt}^2}-e^{-{ipt}^2}dt{pt}^2=ij{pt}^2=-ig$$$$=\frac{1}{2p}{\int }_0^{\mathrm{\infty }}{\left(\frac{ij}{p}\right)}^{-\frac{1}{2}}{e}^{-j}dj+{\left(\frac{-ig}{p}\right)}^{-\frac{1}{2}}{e}^{-g}dg$$$$=\frac{1}{2\sqrt{p}}\mathrm{\Gamma }\left(\frac{1}{2}\right)(e^{\frac{\pi }{4}i}+e^{-\frac{\pi }{4}i})=\sqrt{\frac{\pi }{2p}}$$

Answered by mnjuly1970 last updated on 06/Jan/21

$$solution:\mathrm{\Psi }={\int }_0^{\mathrm{\infty }}\frac{sin\left(px\right)}{x^{\frac{1}{2}}}dx$$$$\mathrm{\Psi }\stackrel{px=\xi }{=}\frac{1}{p^{\frac{1}{2}}}{\int }_0^{\mathrm{\infty }}\frac{sin\left(\xi \right)}{{\xi }^{\frac{1}{2}}}d\xi =$$$$=\frac{1}{p^{\frac{1}{2}}}\ast \frac{\pi }{2\mathrm{\Gamma }\left(\frac{1}{2}\right)sin\left(\frac{\pi }{4}\right)}=\frac{\pi }{\sqrt{p}2\sqrt{\pi }\frac{\sqrt{2}}{2}}$$$$=\frac{\pi }{\sqrt{2p\pi }}=\frac{\sqrt{2p\pi }}{2p}=\sqrt{\frac{\pi }{2p}}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com