Formula:3787

Digital Library of Mathematical Functions ID 10.9.E20

${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left(zt\right)(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2},\ldots}}$

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
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

Resources that cite this formula

No records found.