DLMF:10.9.E5

MaRDI QIDQ3768

• Label: DLMF:10.9.E5

Digital Library of Mathematical Functions ID 10.9.E5

${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$

Symbols List

• ${\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the second 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
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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