  This paper presents the analysis of axisymmetric helical magnetorotational
instability (HMRI) in the inviscid limit, which is relevant for astrophysical
conditions. The inductionless approximation defined by zero magnetic Prandtl
number is adopted to distinguish the HMRI from the standard MRI in the
cylindrical Taylor-Couette flow subject to a helical magnetic field. Using a
Chebyshev collocation method convective and absolute instability thresholds are
computed in terms of the Elsasser number for a fixed ratio of inner and outer
radii \lambda=2 and various ratios of rotation rates and helicities of the
magnetic field. It is found that the extension of self-sustained HMRI modes
beyond the Rayleigh limit does not reach the astrophysically relevant Keplerian
rotation profile not only in the narrow- but also in the finite-gap
approximation. The Keppler limit can be attained only by the convective HMRI
mode provided that the boundaries are perfectly conducting. However, this mode
requires not only a permanent external excitation to be observable but also has
a long axial wave length, which is not compatible with limited thickness of
astrophysical accretion disks.
