Riemann surface of $K_0(z)$

The question concerns the modified Bessel function of the second kind of order zero ($K_0(z)$).

How does its Riemann surface looks like? How can one evaluate its values on the "other" sheet(s)?

• It is of the form $g(z)\log z$, where $g$ is entire (has explicit power series), which answers all your questions. – Alexandre Eremenko Aug 12 '14 at 13:02
• @AlexandreEremenko, I believe this is not true (see the 1-st eq. at functions.wolfram.com/Bessel-TypeFunctions/BesselK/06/01/04/01/… ). However, $z=0$ is a logarithmic branch point. – bcp Aug 12 '14 at 13:29
• Sorry, it is of the form $g(z)\log z+h(z)$ where $g$ and $h$ are entire functions, which answers all your questions. – Alexandre Eremenko Aug 13 '14 at 11:56