Formula:4096

Digital Library of Mathematical Functions ID 10.22.E25

${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)I_{\nu}\left(z\cos\theta\right)(\tan\theta)^{\mu+1}\mathrm{d}\theta=\frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}{2\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}\left(z\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re\nu>\Re\mu>-1}}$

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
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle\tan\NVar{z}}}$: tangent function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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