Formula:6458

Digital Library of Mathematical Functions ID 18.17.E20

${\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)J_{-n-\frac{1}{2}}\left(\tfrac{1}{2}y\right),}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}$: Legendre polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

Resources that cite this formula

No records found.