DLMF:10.32.E13

MaRDI QIDQ4230

• Label: DLMF:10.32.E13

Digital Library of Mathematical Functions ID 10.32.E13

${\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}z)^{-2t}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle c>\max(\Re\nu,0),|\operatorname{ph}z|<\frac{1}{2}\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: 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
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the second kind
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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