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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)?

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  • $\begingroup$ It is of the form $g(z)\log z$, where $g$ is entire (has explicit power series), which answers all your questions. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 12, 2014 at 13:02
  • $\begingroup$ @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. $\endgroup$
    – bcp
    Commented Aug 12, 2014 at 13:29
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    $\begingroup$ Sorry, it is of the form $g(z)\log z+h(z)$ where $g$ and $h$ are entire functions, which answers all your questions. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 13, 2014 at 11:56

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