Question and Answers Forum

All Questions      Topic List

Vector Calculus Questions

Previous in All Question      Next in All Question      

Previous in Vector Calculus      Next in Vector Calculus      

Question Number 210554 by Batmath last updated on 12/Aug/24

Answered by MrGaster last updated on 02/Nov/24

$${\int }_0^{\mathrm{\infty }}[\frac{1}{2}-S\left(px\right)]x^{2q-1}dx={\int }_0^{\mathrm{\infty }}[\frac{1}{2}-\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\sin \left(tpx\right)}{t}dt]x^{2q-1}dx$$$$=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{1}{2}[\frac{1}{2}-\frac{\sin \left(tpx\right)}{t}]x^{2q-1}dtdx$$$$=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{1}{2}{x}^{2q-1}dxdt-\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{\sin \left(tpx\right)}{t}{x}^{2q-1}dtdx$$$$=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{2}\frac{x^{2q}}{2q}{\mid }_0^{\mathrm{\infty }}dt-\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{t}\left[\frac{-\cos \left(tpx\right)}{2qp}{x}^{2q}\right]{\mid }_0^{\mathrm{\infty }}dt$$$$=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{2}\cdot \frac{1}{2a}\underset{x\to \mathrm{\infty }}{\lim }{x}^{2a}dt-\frac{1}{x}{\int }_0^{\mathrm{\infty }}\frac{1}{t}\cdot \frac{1}{2qp}[\underset{x\to \mathrm{\infty }}{\lim cos}\left(tqx\right)x^{2q}-\underset{x\to 0}{\lim cos}\left(tqx\right)x^{2q}]dt$$$$=\frac{1}{\pi }\cdot \frac{1}{2q}\cdot \frac{1}{2}\cdot \mathrm{\infty }-\frac{1}{\pi }\cdot \frac{1}{2qp}[0-1]{\int }_0^{\mathrm{\infty }}\frac{1}{t^{2q+1}}dt$$$$=\frac{1}{\pi }\cdot \frac{1}{2qp}\cdot \frac{\pi }{2}\cdot \frac{1}{\mathrm{\Gamma }(2q+1)}$$$$=\frac{1}{4qp}\cdot \frac{\mathrm{\Gamma }(2q+1)}{\mathrm{\Gamma }(2q+1)}$$$$=\frac{1}{4qp}\cdot \frac{\mathrm{\Gamma }(2q+1)}{2q\mathrm{\Gamma }\left(2q\right)}$$$$=\frac{1}{4qp}\cdot \frac{\mathrm{\Gamma }(2q+1)}{2\sqrt{\pi }\mathrm{\Gamma }(2q+\frac{1}{2})}$$$$=\sqrt{2}\frac{\mathrm{\Gamma }(q+\frac{1}{2})\sin \frac{2q+1}{4}\pi }{4\sqrt{\pi }{qp}^{2q}}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com