DLMF:18.17.E18

MaRDI QIDQ6456

• Label: DLMF:18.17.E18

Digital Library of Mathematical Functions ID 18.17.E18

${\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{\lambda}}.}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)}}$: ultraspherical (or Gegenbauer) polynomial
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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