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We consider the metric of an axially symmetric rotating black hole. We do notspecify the concrete form of a metric and rely on its behavior near the horizononly. Typically, it is characterized (in the coordinates that generalize theBoyer-Lindquist ones) by two integers $p$ and $q$ that enter asymptoticexpansions of the time and radial metric coefficients in the mainapproximation. For given $p,$ $q$ we find a general form for which the metricis regular, and how the expansions of the metric coefficients look like. Wecompare two types of requirement: (i) boundedness of curvature invariants, (ii)boundedness of separate components of the curvature tensor in a free fallingframe. Analysis is done for nonextremal, extremal and ultraextremal horizonsseparately.

Axially symmetric rotating black hole with regular horizons